# Previous speakers

 Jeroen Lamb Krámli András Jon Aaronson Carlangelo Liverani Shigeki Akiyama Stefano Luzzatto Pavel Bacsurin Anthony Manning Viviane Baladi Jens Marklof Bálint Péter Ian Melbourne Oscar Bandtlow Thierry Monteil Bárány Balázs Nándori Péter Vitaly Bergelson Némedy Varga András Borbély Gábor Pajor-Gyulai Zsolt Henk Bruin Paulin Dániel Buczolich Zoltán Francoise Pene Oliver Butterley Harald Posch Jean-Pierre Conze Michal Rams Vitalie Cracan Makiko Sasada Mark Demers Scheuring István Carl Dettmann Klaus Schmidt Dmitry Dolgopyat Pablo Shmerkin Andrew Ferguson Simányi Nándor Bastien Fernandez Simon Károly Garay Barnabás Boris Solomyak Thomas Gilbert Mikko Stenlund Alexander Grigo Sütõ András Halász Miklós Szász Domokos Hatvani László Dalia Terhesiu Horváth Miklós Tél Tamás Thomas Jordan Joerg Thuswaldner Vadim Kaimanovich Mike Todd Fanny Kassel Tóth Bálint György Károlyi Tóth Imre Péter Zemer Kosloff Varjú Tamás Robert T. Kozma Lai-Sang Young Atahualpa S. Kraemer Roland Zweimüller

Jon Aaronson (Tel Aviv University)

Some remarks on dissipative, ergodic, measure preserving transformations.

27. January 2006.

Abstract: A conservative, ergodic, mpt of a sigma-finite measure space, has no other sigma-finite, absolutely continuous, invariant measures other than constant multiples of the original. We show that a dissipative, ergodic mpt of a sigma-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. (Joint work with Tom Meyerovitch.)

Shigeki Akiyama (University of Tsukuba)

Non-periodic expansions of one.

21. March 2014.

Abstract: Usually proving non periodicity of an orbit of a dynamical system is more difficult than periodicity. So in general, it is of interest to give a new idea to show non-periodicity. Here we will prove that the expansion of one of algebraic integers defined by x^n-x-1 for n=4,5, \dots is not periodic. This means that the associated beta expansion gives non-sofic symbolic system. I will talk on some known methods to show non-periodicity of beta expansion, open problems, a new ingredient in the above proof and its generalization.

Pavel Bacsurin

An estimate for a volume of a neighborhood of a zero locus of smooth function

9. December 2005.

Abstract: We show that a d-neighborhood of a level set of a smooth function which never vanishes to infinite order has volume of order d. (joint with Ch. Fefferman)

Pavel Bacsurin

Fundamental theorem for multidimensional dispersing billiards and the structure of singularity manifolds

6. December 2005.

Abstract: Fundamental theorem is an important tool introduced by Sinai in 1970 to prove (local) ergodicity of dispersing billiards. Its proof requires certaint regularity of the sets, where the billiard map is discontinuous. I'll discuss the proof of the fundamental theorem and the structure of these sets in case when the scatterers are finitely smooth.

Anisotropic spaces of distributions for hyperbolic dynamics: transfer operators and dynamical determinants

21. April 2006.

Zsolt Pajor-Gyulai (University of Maryland)

From averaging to homogenization in cellular flows - an exact description of the transition.

19. September 2014.

Abstract: We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained. Joint work with M. Hairer and L. Koralov.

Roland Zweimüller (University of Vienna)

Return- and Hitting-time limits for rare events in infinite measure preserving systems.

19. September 2014.

Abstract: While there exists a large body of work analysing "return- and hitting-time statistics" (limit laws for the return- or hitting times of small sets) for finite measure preserving dynamical systems, very little is known in the infinite measure (null recurrent) case. I will report on joint work with F Pene and B Saussol (Brest, F) which clarifies what to expect in situations with reasonable ergodic/mixing properties. Specific results for dynamical systems concern cylinders of Markov maps. Our considerations also yield results for null-recurrent Markov chains which (somewhat surprisingly) seem to be new.

Péter Bálint (BME MI)

On the work of Nikolai Chernov II.

19. September 2014.

Abstract: On August 7, 2014, the great mathematician, Nikolai Chernov, our close friend passed away. In this seminar we commemorate him by explaining some of his most exceptional ideas.

Bálint Péter

Egy hõvezetési jelenségek által motivált egyszerû dinamikai rendszer (Lai-Sang Younggal közös, folyamatban lévõ kutatásokról)

6. April 2007.

Abstract: "Egy egyszerû egydimenziós modellrõl szeretnék beszélni, amelyben - a lánc két végén elhelyezett sztochasztikus hõtartályoktól eltekintve - az idõfejlõdés determinisztikus. Erre a dinamikára explicit módon felírható egy invariáns mérték, amely
i) nem mutat lokális termikus egyensúlyt és
ii) amelyre nézve a hõmérsékleti profil nem-lineáris.
Jelenleg a kapott invariáns mérték unicitásának bizonyításán dolgozunk. Szeretném röviden vázolni az erre vonatkozó gondolatmenetet is."

Oscar Bandtlow (Queen Mary University of London)

Hölder continuity of the topological entropy for expanding interval maps with holes.

8. May 2014.

Abstract: This talk is on recent work with H.H. Rugh. For piecewise monotonic expanding interval maps with holes we show that, under a non-degeneracy condition on the map, the topological entropy is Hölder continuous with respect to hole position and size. The Hölder exponent is related to the entropy itself and to expansion rates of the map.

Balázs Bárány (University of Warwick)

Ledrappier-Young formula for self-affine measures.

2. April 2015.

Abstract: Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. We show that self-affine measures on the plane satisfy the Ledrappier-Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the so-called dominated splitting. We give a sufficient conditions, inspired by Ledrappier, that the dimensions of such self-affine measure is equal to the Lyapunov dimension. We show some applications, as well.

Bárány Balázs (IMPAN, Varsó)

Iterált függvényrendszerek helyfüggõ valószínûségekkel.

2. April 2013.

Absztrakt: Az elõadás során az Iterált Függvényrendszerek (IFS) invariáns mértékeinek egy kiterjesztésével foglalkozunk, ahol a megfelelõ valószínûségek Hölder-folytonos függvények. Megmutatjuk, ha az IFSek egy paraméterezett családja teljesíti a transzverzalitási feltételt, akkor a mérték dimenziója megegyezik az entrópia és Lyapunov exponens hányadosával, valamint ha ez a hányados nagyobb, mint egy, akkor abszolút folytonos majdnem minden paraméterérték esetén. A fenti eredményt alkalmazható a Bernoulli konvolúció és néhány SBR mérték abszolút folytonosságának vizsgálatához.

Bárány Balázs (BME MI)

Box-dimension of the generalized 4-corner set and its projections.

14. May 2010.

Abstract: In the last two decades considerable attention has been paid to the dimension theory of self-affine sets. In the case of the generalized four corner sets, the IFS obtain as the projection of the self-affine system have maps of common fixed points. We extend our previous result which introduced a new method of computation of the box and Hausdorff dimension of self-similar families where some of the maps have common fixed point. The extended version of our method makes it possible to determine the Box dimension of the generalized four corner set for Lebesgue-typical parameters.

Bárány Balázs (BME MI)

Invariáns mértékek abszolút folytonossága nem affin IFS-re.

25. April 2008.

Vitaly Bergelson (Ohio State University, Columbus)

From Sarkozy's difference theorem to polynomial extensions of Szemeredi's theorem on arithmetic progressions.

6. September 2013.

Abstract: We will start with describing the ergodic counterpart (due to Furstenberg) of A. Sarkozy's theorem which states that for any set S of natural numbers which has positive upper density, there exist x and y in S and a positive integer n, such that x - y = n^2.
We will discuss then various extensions of this theorem including the polynomial multiple recurrence results which are behind the polynomial extensions of the celebrated Szemeredi's theorem on arithmetic progressions.

Borbély Gábor (BME)

Study of Decay of Correlations in a System of Two Falling Balls

28. May 2010.

Abstract: The system of falling balls is a well known dynamical system studied by (among many others) Wojtkowski in the 90's. In my thesis I study this system with two particles and examine some conditions in order to prove decay of correlations with a polynomial rate. In the study of the stochastic properties of a dynamical systems the most crucial point is the decay of correlations. Proving such a slow mixing rate is based on the results of Chernov and Zhang and also uses the results of Young. From Wojtkowski we know that the system is hyperbolic and we introduce a first return map in order to ensure uniformly hyperbolic. One part of the proof is to obtain a tail bound for the distribution of the return times by the analysis of the first return sets.

Henk Bruin (Universität Wien, Austria)

Thermodynamics for Lebesgue dissipative interval maps.

21. September 2012.

Abstract: Using inducing schemes (generalised first return maps) to obtain uniform expansion is a standard tool for (smooth) interval maps, in order to prove, among other things, the existence of invariant measures, their mixing rates and stochastic laws. In this talk I would like to present joint work with Mike Todd (St Andrews) on how this can be applied to maps on the brink of being dissipative. We discuss a family f of Fibonacci maps for which Lebesgue-a.e. point is recurrent or transient depending on the parameter . The main tool is a specifi c induced Markov map F with countably many branches whose lengths converge to zero. Avoiding the difficulties of distortion control by starting with a countably piecewise linear unimodal map, we can identify the transition from conservative to dissipative exactly, and also describe in great detail the impact of this transition on the thermodynamic formalism of the system (existence and uniqueness of equilibrium states, (non)analyticity of the pressure function and phase transitions).

Zoltán Buczolich (ELTE)

Averages along the squares and related topics.

24. May 2013.

Buczolich Zoltán

Számolás és konvergencia az ergodelméletben

4. May 2007.

Abstract

Oliver Butterley (University of Vienna)

The Lorenz flow and some functional analysis.

25. October 2013.

Abstract: I will recall how the Lorenz flow arises from studying convection and what is known of the dynamical properties of it, such as the existence of an attractor and singular hyperbolicity. I will then discuss the progress I have made on the question of rate of mixing, namely establishing the functional-analytic framework for this problem and that the Laplace transform of the correlation function admits a meromorphic extension into a strip about the imaginary axis. Finally I will explain the potential for extending this argument and the difficulties involved.

Jean-Pierre Conze (IRMAR, Université de Rennes)

Limit theorems for some sequential dynamical systems

5. October 2007.

Abstract: A sequential dynamical systems is a sequence $(\theta_n = \tau_n \circ \tau_{n-1} \cdots \circ \tau_1, n \ge 1)$, where $(\tau_n)_{n \ge 1}$ is a sequence of nonsingular maps of a probability space $(X, {\cal A}, m)$ into itself. The extension of notions, like ergodicity or mixing, from the case of the iterates of a single transformation to a sequential system was considered by D. Berend and V. Bergelson in 1984. Some authors gave examples of sequential systems of hyperbolic type: there are results of V.I. Bakhtin (1994); a property of stable mixing for a sequence of automorphisms of the 2-torus was shown by L. Polterovich and Z. Rudnick (2001). The case of sequential systems generated by random sequences of transformations appears also in the literature. For a regular function $f$ on $X$, one can ask if the sums $\sum_{k=1}^n f(\tau_k \tau_{k-1} ... \tau_1 x)$ satisfy limit theorems, like the law of large numbers or the CLT. After discussing some general notions, we present two examples where limit theorems can be obtained (with Albert Raugi for the first one, with Stéphane Le Borgne and Mikaél Roger for the second one) : \hfill \break 1) $(\tau_n)$ is a sequence of expanding maps on $[0,1]$, \hfill \break 2) $\tau_n = A {\rm \ or \ } B$, where $A$ and $B$ are suitable ergodic automorphisms of the torus.

Vitalie Cracan (CEU)

Interval exchange transformations

14. October 2010.

Vitalie Cracan (CEU)

A rotation-reflection system.

28. May 2010.

Abstract: We consider a skew-product on the torus - a rotation "perturbed" by reflection - and ask about the density of the orbits in this system. In the talk I will speak about some possible ways to go about this question, but will not be able to give a satisfactory asnwer. By understanding this simplified rotation-reflection system we hope to to be able to tackle a more complicated one which comes from rotor interaction in the annulus billiard model.

Mark Demers (Fairfield University)

Perturbations of dispersing billiards via spectral methods.

24. May 2013.

Abstract: We will discuss perturbations of the billiard map associated with a periodic Lorentz gas via the stability of the spectrum of the associated transfer operator. Recently, we constructed Banach spaces and norms on which the transfer operator for the unperturbed billiard map enjoys a spectral gap. We will present a number of perturbations which fit into this functional analytic framework and for which the spectral gap persists, including: movements and deformations of scatterers, external forces with thermostatting, twists or kicks at reflections, and random perturbations composed of these various classes. This approach recovers many known results for these systems and establishes several new ones. This is joint work with Hongkun Zhang.

Carl Dettmann (Bristol University)

New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis.

19. January 2012.

Abstract: The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor sqrt(t) are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by sqrt(t ln t), with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.

Carl Dettmann (Bristol University)

Escape from systems with small holes.

17. January 2012.

Abstract: A dynamical system may be "opened" by allowing trajectories to leak out through one or more holes, a subset of phase space. Given a distribution of initial conditions, We can then pose questions about the probability of surviving within the system, as a function of time, the size and position of the hole(s). Open billiard dynamics can be related to a number of physical experiments and applications involving escape of particles from a cavity. In several geometries the leading coefficient of the survival probability can be determined, including connections with the Riemann Hypothesis and phenomena such as asymmetric transport. Recent and detailed results for escape and diffusion in one-dimensional maps will also be discussed.

Dmitry Dolgopyat (University of Maryland, College Park)

Current in periodic Lorentz gas with infinite horizon.

30. June 2009.

Abstract: We study electrical current in two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the voltage difference E, i.e. J = D E/2 + o(E), where $D$ is the diffusion matrix of the Lorentz particle moving freely without electrical field (Chernov-Eyink-Lebowitz-Sinai) This formula agrees with classical Ohm's law and the Einstein relation. Here we investigate the more difficult model with infinite horizon. We find that infinite corridors between scatterers allow the particles (electrons) move faster resulting in an abnormal current (causing superconductivity'). Precisely, the current is now given by J = (D/2) E |log E| +O(E), where D is the superdiffusion' matrix of the Lorentz particle moving freely without electrical field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. We also obtain new results for the infinite horizon Lorentz gas without external fields, complementing recent studies by Szász and Varjú.

Andrew Ferguson (University of Bristol)

Dimension of the projections and slices of dynamically defined sets.

23. March 2012.

Abstract: In 1954 Marstrand proved several theorems that concern the almost sure dimension of a projection and slice of a planar Borel set. Later, Kaufman gave bounds on the set of directions for which the projection is exceptional. In this talk I will discuss the situation when one considers certain dynamically defined sets, showing that often one may say quite a bit more.

Andrew Ferguson (University of Bristol)

The dimension of some sets generated by systems with holes

21. October 2011.

Abstract: Let $(X,d)$ denote a compact metric space and $T:X\to X$ an open and expanding map. For a non-empty open set $U\subset X$ we define the survivor set $X_U$ to be the set of points, which under forward iteration do not enter $U$. In this talk I will discuss the dimension theory of these sets for some conformal and non-conformal systems.

Bastien Fernandez (Centre de Physique Théorique, CNRS Marseille)

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps.

16. May 2014.

Abstract: To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this talk, I will consider a family of CML composed by piecewise expanding individual maps, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. I will show, mathematically for N=3 and numerically for N>3, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extent, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.

Garay Barnabás (BME MI)

Három, mérnököktõl vagy legalábbis nem "tisztán matematikusok"-tól (Chua, Csikja, Hubbard) származó probléma a dinamikai rendszerek körében.

16. May 2008.

Thomas Gilbert (Brussels)

Multi-state Lévy walks to describe the kinetics of diffusive processes combining ballistic motion.

24. April 2015.

Abstract: I will describe a class of Lévy walks on regular lattices specified by continuous time random walks which combine diffusive scattering and ballistic propagation, such that successive renewal events are separated by exponentially-distributed waiting times. Such processes will be analysed in terms of multiple states whose time-evolution obeys a set of coupled delay differential equations. This framework yields a simple derivation of the scaling regimes of the mean squared displacement. Of particular interest is the application of this formalism to infinite horizon billiard models in the limit of narrow corridors, such that a normal contribution to the finite-time diffusion coefficient may become arbitrarily larger than the anomalous component.

Thomas Gilbert (ULB, Brussels)

Order statistics and the Lyapunov spectra of some classes of high-dimensional billiard systems.

16. September 2011.

Abstract: Consider a system made out of a possibly large number of identical copies of a two-dimensional dispersive billiard table and let us further assume a form of infrequent pairwise energy-preserving interaction among them. The interaction we will consider will typically be of collisional type and may therefore induce the exchange of a substantial amount of energy among the colliding pair. The question we address is the following: What is the spectrum of Lyapunov exponents of such a system? It turns out this question is closely related to a famous problem in probability theory, first addressed by Laplace in his attempt to construct an error function towards the end of the 18th century: What is the distribution of the ordered lengths of a fixed number of random divisions of the unit interval?

Alexander Grigo (University of Oklahoma)

Elliptic periodic orbits in C^2-smooth stadium billiards.

11. January 2013.

Abstract: The stadium billiard is one of the simplest examples of hyperbolic and ergodic convex billiards. This came as a great surprise, because Lazutkin showed in 1973 that strictly convex billiard tables with smooth enough boundary have caustics, hence cannot be ergodic. When smoothing out the ends of the circular arcs of the usual stadium billiard such that the curvature of the resulting curved segment vanishes at its endpoint one obtains a $C^2$-smooth stadium. We show that even for arbitrarily short smoothening regions the resulting $C^2$-smooth stadium billiard has elliptic periodic orbits for arbitrary short and also for arbitrary large separation distances of the two curved boundary components. In particular, this result shows that the billiard dynamics on a table with focusing boundary components can be dramatically affected by changing the smoothness of the boundary arbitrarily close to just a single point. (This is joint work with Leonid Bunimovich.)

Alexander Grigo (University of Toronto)

Hydrodynamic limits of the inelastic Boltzmann equation via invariant manifolds.

11. June 2010.

Abstract: The Boltzmann equation is a very efficient model to describe the evolution of gas-like systems. In this talk we consider interactions that dissipate energy, and will present several methods to deduce hydrodynamic equations for the macroscopic description. After briefly reviewing scaling limits and series expansion techniques (which are well studied in the literature) we present an approach based on dynamical systems techniques, which make it possible to relate the dynamics of the Boltzmann equation to a reduced dynamics on a slow manifold''. For conservative interactions this slow dynamics reduces to the Navier-Stokes equations. The main emphasis will be on how this dynamical system approach allows to relate various old and new results on hydrodynamic limits within one framework. This is joint work with Eric Carlen and Shui-Nee Chow.

Halász Miklós

Ergodicitás és szigetek egy két paraméteres biliárd-családban

28. June 2006.

Hatvani László (SZTE Bolyai Intézet)

Véletlen együtthatós másodrendû lineáris differenciálegyenletek stabilitási tulajdonságai

14. December 2007.

Abstract: Az x'' + a^2(t)x=0 egyenletet nézzük, ahol a olyan lépcsõs függvény, amelyben a lépcsõk magasságainak {a_k} sorozata adott, viszont az ugráshelyek {t_k} sorozata véletlen abban az értelemben, hogy t_k-t_{k-1} különbségek egymástól független , pozitív értékû, nem szükségképpen azonos eloszlású valószínûségi változók. Két alapvetõ esetet vizsgálunk:
1. Az {a_k} sorozat monoton növekedve végtelenhez tart. Belátjuk, hogy igen általános feltételek mellett majdnem biztosan az egyenlet minden megoldására lim x(t)=0 teljesül (stabilitás).
2. Az {a_k} sorozat két, egymáshoz közeli számból áll és periodikus, valamint a t_k-t_{k-1} különbségek egymástól független pozitív értékû, azonos eloszlású valószínûségi változók (Meissner-egyenlet, a hintázás problémája). Feltételeket adunk a t_k-t_{k-1} valószínûségi változó karakterisztikus függvényére, amelyek teljesülése esetén a teljes mechanikai energia várható értékei végtelenhez tartanak, ha k tart végtelenhez (instabilitás).

Horváth Miklós (BME MI)

7. March 2008.

Abstract: A vizsgálandó kérdés a Sturm-Liouville operátor felismerése és rekonstruálása sajátértékeibõl. A hangsúly az unicitáson lesz, de beszélünk a feladat megoldhatóságáról, a megoldás stabilitásáról, megoldási algoritmusokról és mindezek kapcsolatáról egy inverz szórási feladattal.

Thomas Jordan (University of Bristol)

Multifractal analysis for quotients of Birkhoff sums for countable Markov maps.

5. April 2013.

Abstract: We consider the multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the considerable differences between this setting and the uniformly hyperbolic setting. This is joint work with Godofredo Iommi.

Stopping times and Poisson boundaries.

24. April 2015.

Abstract: The Poisson boundary of a random walk on a group is defined as the space of ergodic components of the time shift in its path space. In this talk I will discuss how Markov stopping times applied to the path space (in a way similar to time changes in the classical dynamical setup) give rise to new random walks with the same Poisson boundary.

Fanny Kassel

Equidistribution of integral lattices of covolume N

23. February 2007.

Abstract: The subject of equidistribution of finite number of points in homogeneous spaces is a subject in great expansion, combining dynamical systems, group theory, and number theory. I will deal with the following example. Consider the set X of lattices of R^n up to homothety. It is a homogeneous space, naturally endowed with a probability measure. I will show that the sets of sublattices of Z^n of covolume N get equidistributed in X as N tends to infinity. This means that for every compact subset C of X that is "regular" enough (i.e. whose boundary has measure zero), the number of lattices of covolume N that are in C is asymptotically equal to the measure of C.

György Károlyi (BME NTI)

Doubly transient chaos in autonomous dissipative systems.

28. November 2014.

Abstract: Chaos is traditionally studied for trajectories that are either permanently erratic or transiently influenced by permanently erratic ones lying on a set of measure zero. The latter gives rise to the final state sensitivity observed in connection with fractal basin boundaries in conservative scattering systems and driven dissipative systems. We focus on undriven dissipative systems, whose transient dynamics fall outside the scope of previous studies since no time-dependent solutions can exist for asymptotically long times. We show that such systems can exhibit positive finite-time Lyapunov exponents and fractal-like basin boundaries which nevertheless have codimension one. In sharp contrast to its driven and conservative counterparts, the settling rate to the fixed-point attractors grows exponentially in time, meaning that the fraction of trajectories away from the attractors decays superexponentially. While no invariant chaotic sets exist in such cases, the irregular behavior is governed by transient interactions with transient chaotic saddles, which act as effective, time-varying chaotic sets. This is a joint work with Tamás Tél, Márton Gruiz and Adilson Motter.

Zemer Kosloff (Warwick)

Conservative Anosov transformations on the two torus without absolutely continuous Lebesgue invariant measures.

24. April 2015.

Abstract: here

Robert T. Kozma (Stony Brook University)

Limiting Behavior of Julia Sets of Perturbed Quadratic Maps.

27. September 2013.

Abstract: Even the simplest nonlinear function z^2+c gives rise to chaotic dynamics resulting in very intricate mathematical objects such as the Mandelbrot set, and Julia sets. In this talk we first go over some basic properties of the Mandelbrot set, and Julia sets. We then go on to describe novel results on the behavior of Julia sets of certain families of rational maps arising from z^2+c by adding a small perturbation term \lambda/z^2 and taking the limit as \lambda -> 0. We will see that for certain c values the resulting Julia sets have some astonishing geometric and topological properties. Using symbolic dynamics and Cantor necklaces, we prove that as \lambda -> 0, the Julia for this family set evolves into a space-filling fractal curve.

Atahualpa S. Kraemer (Dusseldorf)

On the free path lengths distribution for quasiperiodic Lorentz gases.

27. February 2015.

Abstract: We will use a construction (based on the projection method) that "periodizes" a quasiperiodic lattice of obstacles, i.e., embed it into a unit cell in a higher-dimensional space, to study the free path lengths distribution for quasiperiodic Lorentz gases. This construction allows us to simulate the dynamics of particles in quasiperiodic structures and to show the generic existence of channels, where particles travel without colliding, up to a critical obstacle radius. The existence of channels depends on the radius of the obstacles, this let us to pass from a system with finite horizon to one with infinite horizon by changing the size of obstacles. We will prove that for the critical radius there is not an upper bound for the free path lengths, but any infinity line (except maybe one) intersects infinite obstacles. We will also show with heuristic arguments and simulations that the free flight distribution in this case is different to both, periodic and random Lorentz gases.

Krámli András (MTA SZTAKI + Szeged)

Nemegyensúlyi energia profil (L.-S. Young - Eckmann, és Levin K. Lin - L.-S. Young alapján)

14. January 2011.

Abstract: A modell 1 dimenziós: kis résekkel egymáshoz ragasztott négyzetbõl kivágott körcikkekbõl álló N darab biliárd asztal képezi a modell alapját. A bal és jobb széleken hõtartályok vannak, amelyek Poisson folyamat szerint (ezek intenzitása lehet más a jobb és bal széleken) pumpálnak be pontszerû de véges (1) tömegû részecskéket a rendszerbe, melyek sebességeloszlása nem nyilvánvaló módon függ a hõmérséklettõl. Ugyanakkor nyomtalanul elnyelik a résen távozó részecskéket. A részecskék közvetlenül nem hatnak egymásra, de minden cella közepén van egy rögzített tengelyû szabadon forgó korong, amellyel a részecskék ütközhetnek. Az ütközésnél a korongra merõleges sebesség komponens elõjelet vált, míg az érintõleges komponens sebességet cserél a koronggal: azaz a korong szögsebessége megkapja a részecske érintõleges sebességét, míg az új érintõleges sebesség a korong korábbi szögsebessége lesz (ez adott tömegû részecskék esetén a korong átmérõjének és tehetetlenségi nyomatékának alkalmas megválasztásával realizálható úgy, hogy az energiamegmaradás ne sérüljön). A részecskesûrûség és energiasûrûség profil lineáris lesz, tehát a hõmérséklet profil törtlineáris.

Krámli András (MTA SZTAKI + Szeged)

Hõvezetés a Lorentz gáz egy karikatúramodelljében.

10. December 2010.

Abstract: A címben megadott karikatúra modell a belsõ állapotú bolyongás [véges sok állapottal], az ugrások közötti belsõ állapot változások felelnek meg a részecske ütközési paraméterei megváltozásának. Az ún. lokális termikus egyensúly nélküli modellben a szórótestek nem vesznek részt a hõátadásban, viszont két típusú [piros:= meleg, kék:= hideg] részecskék bolyonganak két hõtartály között, a hõtartályokról történõ visszatérés során változhat a típus. Kiszámoljuk az energia transszport aszimptotikus viselkedését, és a hõmérséklet profilt diffúziós limeszben midõn a hõtartók távolsága (L) tart a végtelenhez. A hõmérséklet profil törtlineáris lesz. A lokális termikus egyensúly esetén csak egyféle részecske bolyong a szórótestek stacionáris eloszlás szerinti hõmérséklete azonos a bolyongó részecskék ugyanezen eloszlás szerinti hõmérsékletével; itt a hõmérséklet profil lineáris lesz. Módszer: Keldis elmélete a mátrixpolinomokról.

Jeroen Lamb (Imperial College London)

Dynamics of Coupled Maps in Heterogeneous Random Networks.

17. December 2013.

Abstract: We study expanding circle maps interacting in a heterogeneous random network. Heterogeneity means that some nodes in the network are massively connected, while the remaining nodes are only poorly connected. We provide a probabilistic approach which enables us to describe the effective dynamics of the massively connected nodes when taking a weak interaction limit. More precisely, we show that for almost every random network and almost all initial conditions the high dimensional network governing the dynamics of the massively connected nodes can be reduced to a few macroscopic equations. Such reduction is intimately related to the ergodic properties of the expanding maps. This reduction allows one to explore the coherent properties of the network. Joint work with Tiago Pereira and Sebastian van Strien (both at Imperial College).

Carlangelo Liverani (Universita di Roma Tor Vergata)

Fine Statistical Properties in a Class of Two-dimensional Piecewise Hyperbolic Maps (from a joint work with Mark F. Demers)

27. April 2006.

Stefano Luzzatto (ICTP Trieste)

SRB measures for partially hyperbolic systems whose central direction is weakly expanding.

27. February 2015.

Abstract: We consider partially hyperbolic $$C^{1+}$$ diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition $$E^s\oplus E^{cu}$$. Assuming the existence of a set of positive Lebesgue measure on which $$f$$ satisfies a weak nonuniform expansivity assumption in the centre~unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs-Markov-Young geometric structure with integrable return times. We also characterize in this setting SRB measures which are liftable to Gibbs-Markov-Young structures.

Anthony Manning (Warwick)

Curves of fixed points of trace maps.

11. April 2008.

Abstract: We study certain diffeomorphisms of R^3 (where the coordinates are traces of 2x2 matrices) induced by elements of PSL(2,Z). These diffeomorphisms preserve certain level surfaces where the maps are area-preserving with hyperbolic or elliptic behaviour. One such surface is a quotient of a torus and the action is by automorphisms. Hyperbolic fixed points persist at nearby levels and we explain in certain cases where such curves of fixed points return to that level. This work is joint with S Humphries and appeared in ETDS 2007.

Anthony Manning

The volume entropy of a compact surface decreases along the Ricci flow.

14. September 2007.

Abstract: Volume entropy measures the exponential growth of volume/area in the universal cover of a compact Riemannian manifold. On a surface, the Ricci flow changes the metric towards one of constant curvature. We shall discuss these ideas and show that, when the Ricci flow starts at a metric of variable negative curvature, the volume entropy decreases.

Jens Marklof (Bristol)

Escape from the circle

27. March 2009.

Abstract: Bunimovich and Dettmann recently studied the escape rate for a circular billiard with small holes of size h, in the limit of long times t >> 1/h, and found a striking connection with the Riemann hypothesis. In this lecture I report on recent work (in progress) on the limiting distribution for the escape rate with h t fixed, which was observed numerically by Bunimovich and Dettmann. The techniques are closely related to my joint work with Strombergsson on the Boltzmann-Grad limit of the periodic Lorentz gas.

Ian Melbourne (University of Warwick)

Mixing and rates of mixing for infinite measure systems with discrete and continuous time.

30. November 2012.

Abstract: A recent paper, joint with Dalia Terhesiu, developed a theory of mixing and rates of mixing for discrete dynamical systems with an infinite invariant measure. The method combines operator renewal theory techniques (Sarig, Gouezel) with probabilistic ideas of Garsia & Lamperti.
In this talk, I will review these results. Also, I will describe some recent work, again joint with Dalia Terhesiu, where we obtain similar results for continuous time. The techniques are similar, but the estimates are trickier than for discrete time and ideas of Dolgopyat are also required.

Ian Melbourne

Decay of correlations for Lorentz gases

16. November 2007.

Abstract: In this talk, I will describe recent, and in some cases on-going, results on decay of correlations for various continuous time Lorentz gas models, including (in)finite horizon Lorentz gases, Bunimovich stadia, and cuspoidal domains.

Thierry Monteil (Marseille)

Finite blocking property in polygonal billiards and translation surfaces

24. February 2006.

Abstract: A planar polygonal billiard (resp. a translation surface) P is said to have the finite blocking property if for every pair (O,A) of points in P there exists a finite number of "blocking" points B_1,...,B_n such that every trajectory from O to A meets one of the B_i's. The talk will focus on this property, that can be considered as a property about illumination in P. We will relate it to the geometry of the surface (being a flat torus branched covering) and a property of the directional flow on the surface (pure periodicity). The equivalence between those three notions holds under a homological condition on P. Also, we will give a complete classification for Veech surfaces and the surfaces of genus 2.

Nándori Péter (Courant Institute, New York University)

Local thermodynamic equilibrium for stochastic models of heat conduction.

1. July 2014.

Abstract: We consider a class of stochastic models that are realizations of Hamiltonian models of heat conduction. By using an extended notion of duality (i.e. conditional and partial duality), we prove local and mesoscopic thermodynamic equilibrium with harmonic energy profile. These results generalize earlier work of Ravishankar and Young by considering arbitrary dimension and arbitrary density of particles. The talk is based on a joint work in progress with Y Li and L-S Young.

Nándori Péter (BME MI)

Lorentz Process with shrinking holes in a wall.

1. March 2013.

Abstract: We ascertain the diffusively scaled limit of a periodic Lorentz process in a strip with an almost reflecting wall at the origin. Here, almost reflecting means that the wall contains a small hole, which is shrinking in time. The limiting process is a quasi-reflected Brownian motion, which is Markovian but not strong Markovian. Local time results for the periodic Lorentz process, having independent interest, are also found and used. Finally, we mention some possible extensions. This is a joint work with Domokos Szasz.

Nándori Péter (BME MI)

Polygonal billiards and the Teichmuller geodesic flow on moduli spaces.

13. April 2012.

Abstract: The celebrated theorem of Kerckhoff, Masur and Smillie (Annals of Maths. '86) states that in a polygon, the angles of which are rational multiples of pi, the billiard flow in almost all directions is uniquely ergodic. The proof uses the beautiful and quite involved construction of the Teichmuller geodesic flow on moduli spaces of Riemann surfaces. Here, we want to understand as much as possible from this theory. The talk is based on a mini course given by Sebastien Gouezel in Warwick, December 2011.

Nándori Péter (BME MI)

S. Sethuraman, S.R.S. Varadhan: A martingale proof of Dobrushin's theorem for non-homogeneous Markov chains.

11. March 2011.

Abstract of the paper: In 1956, Dobrushin proved a definitive central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given through martingale approximation.

Nándori Péter (BME MI)

Recurrence properties of a special type of Heavy-Tailed Random Walk.

17. September 2010.

Abstract: In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate result was needed concerning the recurrence properties of its unperturbed version. These were analogous to the similar properties of Simple Symmetric Random Walk. However, in the case of Lorentz process with infinite horizon, the analogous results for the corresponding random walk are not known, either. In this talk, these properties are ascertained for the appropriate random walk (this happens to be in the non normal domain of attraction of the normal law). As a tool, an estimation of the remainder term in the corresponding local limit theorem is computed.

Nándori Péter (BME MI)

Elõadás sorozat: Standard pairs, coupling, correlations and stochastic properties.

12., 19., 26. February and 5. March 2010.

Abstract: Standard pairs were introduced to handle complex issues arising in billiards, but the lectures are targeted to the wider audience, the method seems to be generally applicable. The lectures will follow N. Chernov: Advanced statistical properties of dispersing billiards J. of Stat. Phys., 122 (2006), 1061-1094., Elérhetõ az interneten itt A new approach to statistical properties of hyperbolic dynamical systems emerged recently; it was introduced by L.-S. Young and modified by D. Dolgopyat. It is based on coupling method borrowed from probability theory. We apply it here to one of the most physically interesting models - Sinai billiards. It allows us to derive a series of new results, as well as make significant improvements in the existing results. First we establish sharp bounds on correlations (including multiple correlations). Then we use our correlation bounds to obtain the central limit theorem (CLT), the almost sure invariance principle (ASIP), the law of iterated logarithms, and integral tests.

Nándori Péter (BME MI)

Belsõ állapotú bolyongások

17. October 2008.

Abstract: A belsõ állapotú bolyongást, mint modellt, (továbbiakban RWwIS, a Random Walk with Internal States angol elnevezés rövidítésébõl) Sinai definiálta 1981-ben. Eredeti célja az volt, hogy a matematikailag nehezebben kezelhetõ Lorentz folyamatot közelíthetõvé tegye RWwIS-ek segítségével (a belsõ állapotok a Markov felbontás elemei!). Kiderült, hogy a RWwIS önmagában is érdekes, sõt, más alkalmazásai is találhatók, például sorbanállási rendszerek vizsgálatánál. A modell az egyszerû szimmetrikus véletlen bolyongás általánosítása. RWwIS esetén a d dimenziós kockarácson bolyongórészecske egyes lépései általában nem függetlenek egymástól, hanem a belsõ állapotok Markov láncán keresztül összefüggenek. Ezáltal a jelenségek szélesebb köre vizsgálható, viszont természetesen a vizsgálati módszerek is nehezebbekké válnak. Ennek az elõadásnak fõ célja, hogy a Dvoretzky - Erdõs klasszikus, a közönséges bolyongásokra vonatkozó cikkében található, a meglátogatott pontok számára vonatkozó tételek RWwIS-re történõ általánosítását bemutassa. Elsõ eredményünk szerint magas dimenzió esetén (d > 2) az n ideig meglátogatott pontok számának várható értéke, ahogy azt a klasszikus esetre vonatkozó Pólya tétel is sugallja, sok, azaz n-ben lineáris. Az érdekes eset a két dimenzió. Második eredményünk szerint ekkor ez az érték aszimptotikusan c*n/(logn) - ahogy a Dvoretzky - Erdõs cikkben is - azonban c függ a konkrét RWwIS paramétereitõl. Ezeken az eredményeken kívül becslést adunk az n ideig meglátogatott pontok számának szórásnégyzetére is, ami által bizonyítani tudunk nagy számok gyenge, illetve erõs törvényét. Az elõadásban érintünk olyan kérdéseket is, amelyek nem értelmesek egyszerû szimmetrikus véletlen bolyongásra, viszont Lorentz folyamatra és RWwIS-re igen.

András Némedy Varga (MTA Alfréd Rényi Institute of Mathematics)

High dimensional generalization of standard pairs and the coupling technique.

31. January 2014.

Abstract: For uniformly hyperbolic dynamical systems with singularities one particular technique to prove exponential decay of correlations is the coupling of standard pairs. It was developed by Chernov and Dolgopyat for systems with two dimensional phase spaces. The key idea is the following. Consider any two standard pairs, which are just unstable curves with some measures on them that have sufficiently regular densities. Iterating them forward by the dynamics at certain times some parts of their images will be very close to each other, so that they will be connected by stable manifolds and hence the distance between them converges to zero exponentially fast. At these times - due to the previous reason - the measures they carry may be coupled along stable manifolds. If each time a fix amount of the measures can be coupled and the measure of those points, who are not coupled by time n is exponentially small in n, then the system enjoys exponential decay of correlations and also some other statistical properties (e.g. limit theorems) hold. In this talk I would like to present the high dimensional generalization of this method, enlightening the difficulties arising from the d > 2 setup. This is joint work with Péter Bálint.

Némedy Varga András (BME MI)

High dimensional generalization of standard pairs and the coupling technique.

12. October 2012.

Abstract: For uniformly hyperbolic dynamical systems with singularities one particular technique to prove exponential decay of correlations is the coupling of standard pairs. It was developed by Chernov and Dolgopyat for systems with two dimensional phase spaces. The key idea is the following. Consider any two standard pairs, which are just unstable curves with some measures on them that have sufficiently regular densities. Iterating them forward by the dynamics at certain times some parts of their images will be very close to each other, so that they will be connected by stable manifolds and hence the distance between them converges to zero exponentially fast. At these times - due to the previous reason - the measures they carry may be coupled along stable manifolds. If each time a fix amount of the measures can be coupled and the measure of those points, who are not coupled by time n is exponentially small in n, then the system enjoys exponential decay of correlations and also some other statistical properties (e.g. limit theorems) hold. In this talk I would like to present the high dimensional generalization of this method, enlightening the difficulties arising from the d > 2 setup. This is joint work with Péter Bálint.

Némedy Varga András (BME MI)

Statistical properties of the system of two falling balls

28. October 2011.

Abstract: The system of two falling balls is considered, a physical model first introduced by Wojtkowski, who also identified the cases when the dynamics are hyperbolic and ergodic. For an open set of the external parameter (the ratio of the two masses), we prove polinomial decay of correlations (with logarithmic corrections) and the central limit theorem for a natural class of observables. The proof uses the tower method of Young (in its Chernov-Zhang version) and relies on a detailed geometric analysis of the dynamics, including the expansion and regularity properties of unstable curves, as well as the description of the singularity structure. Joint work with Gábor Borbély and Péter Bálint.

Némedy Varga András (BME MI)

Young tower construction for the singular CAT map.

21. May 2010.

Abstract: The importance of hyperbolic dynamical systems with singularities is well-known in various branches of mathematics and physics. The aim of the studies is to show quasi-randomness and to discuss statistical properties of such a system including decay of correlations and limit theorems. The study of smooth uniformly hyperbolic dynamical systems dates back to the 70's, when Sinai, Ruelle and Bowen proved exponential decay of correlations on them, using the method of Markov partitions. However their proof does not work in presence of singularities. Very recently promising new methods have been developed. A technique that has turned out to be particularly powerful is the tower construction method introduced by Young. To understand how this complicated method works it is worth implementing it on simple toy models of hyperbolic systems with singularities. One particular example is the singular CAT map. It is like the linear automorphism of the torus, but we take the topology of the unit square, instead of the topology of the torus. This will cause some singularities, so we are able to present how the tower construction method of Young works, and prove exponential decay of correlations and central limit theorem on our system.The importance of hyperbolic dynamical systems with singularities is well-known in various branches of mathematics and physics. The aim of the studies is to show quasi-randomness and to discuss statistical properties of such a system including decay of correlations and limit theorems. The study of smooth uniformly hyperbolic dynamical systems dates back to the 70's, when Sinai, Ruelle and Bowen proved exponential decay of correlations on them, using the method of Markov partitions. However their proof does not work in presence of singularities. Very recently promising new methods have been developed. A technique that has turned out to be particularly powerful is the tower construction method introduced by Young. To understand how this complicated method works it is worth implementing it on simple toy models of hyperbolic systems with singularities. One particular example is the singular CAT map. It is like the linear automorphism of the torus, but we take the topology of the unit square, instead of the topology of the torus. This will cause some singularities, so we are able to present how the tower construction method of Young works, and prove exponential decay of correlations and central limit theorem on our system.

Zsolt Pajor-Gyulai (University of Maryland)

From averaging to homogenization in cellular flows - an exact description of the transition.

19. September 2014.

Abstract: We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained. Joint work with M. Hairer and L. Koralov.

Zsolt Pajor-Gyulai (University of Maryland)

A brief look at stochastic averaging and the Freidlin-Wentzell theory.

12. September 2014.

Abstract: The averaging principle, initially formulated by N. N. Bogolyubov in 1945, grew to be one of the fundamental methods of asymptotic analysis of deterministic dynamical systems. Its stochastic counterpart was introduced by R. Z. Khasminskii in 1966, was greatly elaborated in the works of M. I. Freidlin and A. D. Wentzell (and others) and since obtained similar importance in the investigation of stochastic asymptotic problems. The purpose of this talk is to give an exhaustive - although admittedly incomplete - account of the most important milestones of this development.

Zsolt Pajor-Gyulai (University of Maryland, College Park)

An exact description of the transition from averaging to homogenization in cellular flows.

10. January 2014.

Abstract: We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.

Pajor-Gyulai Zsolt (BME FI)

Scaled Type Markovian Renewal Processes with Infinite Mean

2. April 2010.

Abstract: Markovian renewal processes have been widely used tools in applied mathematics for a variety of applications ranging from problems in biology to the risk assessment of seismic disasters. They are generalizations of ordinary renewal processes in the sense that the waiting times have different distributions. However, the type of waiting times changes from one renewal to another according to a Markov chain. Although they have rich literature, no one seems to have treated the case when the means of the waiting times can be infinite. In the talk, some recently discovered result will be presented with the assumption that the distributions of the waiting times are just rescaled copies of a common ancestor distribution.

Pajor-Gyulai Zsolt (BME MI)

A kétrészecskés Lorentz folyamat modellezése belsõ állapotú bolyongással

31. October 2008.

Abstract: A Lorentz folyamat egy régóta vizsgált problémája a dinamikai rendszerek témakörének. Ez az egyszerû konstrukció, melynek viselkedését különbözõ technikákkal próbálták leírni, ezek közül egy a Markov felbontás. Ennek kapcsán vezette be Yakov Sinai a belsõ állapotú bolyongást (angol nevén Random Walk with Internal Degrees of Freedom-RWwIDS) 1981-ben, ötlete szerint a felbontás elemei a bolyongáshoz tartozó belsõ állapotok voltak. A belsõ állapotú bolyongás a közönséges bolyongás általánosítása, a folyamat kiegészül egy belsõ szabadsági fokkal, amelytõl a lépések eloszlása nagyban függ. A kezelhetõség szempontjából fontos kikötés a térbeli transzlációs invariancia, mivel ennek következményeként a belsõ állapotok Markov-láncot alkotnak. Erre a konstrukcióra ismeretes lokális határeloszlástétel. Jelen dolgozat a következõ rendszert fogja vizsgálni. Adott a síkban két Lorentz részecske, melyek a periodikus szórótestek által definiált cellákban vándorolnak különbözõ energiákkal (sebességekkel). Ha a két részecske egy cellába kerül, akkor ott ütközés, ezáltal energiacsere lehetséges a klasszikus mechanika törvényei szerint. Ha azonosítjuk a bolyongó részecske sebességének valamilyen rögzített tengellyel bezárt szögét a belsõ állapotokkal, akkor adja magát az analógia. A feladat megkívánja, hogy folytonos idejû modellt használjunk, ehhez összetett Poisson folyamatot csinálunk a diszkrét bolyongásból. Az energiacsere leírására a klasszikus kemény golyós modellt használjuk. Mivel a két részecske találkozása rendkívül ritka esemény, ezért alkalmazzuk a szokásos molekuláris káoszfeltevést, azaz jelen esetben a belsõ állapotok stacionárius eloszlása egyenletes, az átmenetmagot elég ezen feltétel mellett megadni. Fontos részét képezi a vizsgálatnak a szokásos felújítási elmélet általánosítása arra az esetre, amikor a folyamat csak egy paramétertõl eltekintve tér vissza a kiindulási állapotba. Jelen esetben a visszatérési idõk fognak függeni a részecskék energiáitól, mint paramétertõl, továbbá ezek várható értéke végtelen. Ennek következménye, hogy a maximális kirándulás dominálja a folyamatot, amibõl adódik, hogy a határeloszlás a különbözõ energiák esetén érvényes határeloszlások keveréke valamilyen súlyfüggvénnyel. Lényeges felhasznált irodalom (az alapvetõ valószínûségszámítási irodalom mellett): 1. Krámli - Szász: Random Walks with integral degrees of freedom, Zeitschrift für Wahrscheinlichkeitstheorie verw. Gebiete, 63, 85-95 (1983) 2. Heusler - Mason: On the Asymptotic Behaviour of Sums of Order Statistics from a Distribution with a Slowly varying upper tail, 355-375

Dániel Paulin (University of Singapore)

Efron-Stein-type concentration inequalities by Stein's method for exchangeable pairs.

5. December 2014.

Abstract: The Efron-Stein inequality, and it's extensions to moment bounds and concentration inequalities are important results in probability theory, since they imply most of the known tail bounds for independent random variables. They were proven using the entropy method by Boucheron, Lugosi and Massart. In this talk, we are going to sketch a new proof of these results based on Stein's method for exchangeable pairs.

Dániel Paulin (National University of Singapore)

Koncentrációs egyenlõtlenségek összefüggõ terekben.

8. January 2013.

Abstract: A koncentrációs egyenlõtlenségek nem-aszimptotikus becslést adnak valószínûségi változók egy függvényének a várható értéktõl való eltérésére (tipikus alakjuk (|f(X_1,...,X_n)-E f|>t) A 80-as évek vége óta számos területen (kombinatorika, informatika, stb.) elterjedt az alkalmazásuk. Az elsõ eredmények független változókra vonatkoztak, késõbb különféle összefüggõségei struktúrák, és különféle tulajdonságú függvények mellett is beláttak ilyen becsléseket. Marton Katalin többek között Markov láncokra bizonyított koncentrációs egyenlõtlenségeket. Az elõadás során bemutatom új erdedményeimet Markov láncokra, Dobrushin kondíció esetén (magas hõmérsékletû statisztikus fizikai modellek), és lokális összefüggõségre. Többféle alkalmazás is sorra kerül, pl. MCMC algoritmusok hibabecslése.

Paulin Dániel (National University of Singapore)

Mcdiarmid koncentrációs egyenlõtlensége, és kapcsolata a Markov láncokkal.

16. December 2011.

Absztrakt: Mcdiarmid koncentrációs egyenlõtlensége:
Legyenek X_1,...,X_n független valószínûségi változók, X=(X_1,...,X_n), f: R^n ->R függvény amire teljesül hogy |f(x)-f(y)|<=c_i ha x és y csak az i. koordinátában tér el, akkor

P(f(X)>E(f(X))+t)<=exp(-t^2/(sum c_i^2))
P(E(f(X))-t>f(X))<=exp(-t^2/(sum c_i^2))

Sourav Chatterjee bizonyítása felcserélhetõ valószínûségi változókra és Markov láncokra épül, és általánosítható olyan esetekre is, amikor X_1,...,X_n összefüggõ.

Francoise Pene (Universite' de Bretagne, Brest)

Random walk in a stationary scenery and random walk with stationary orientations

27. April 2007.

Francoise Pene (Universite' de Bretagne, Brest)

Rate of convergence in the central limit theorem for some hyperbolic or partially hyperbolic dynamical systems

25. November 2005.

Abstract: We explain how general theorems can be applied to establish a rate of convergence for some dynamical systems, using their hyperbolic properties. We give examples of systems to which it can be applied. We have to establish some decorrelation property. To do this, we use the following idea : a control of the covariance in which appears separately the regularity along the stable-central direction and the regularity along the unstable direction. Moreover, we explain how to get a Berry-Esseen result for the billiard system with finite horizon. The proof of this result uses the construction of the Young towers and Fourier calculations.

Harald Posch

Tangent-space dynamics of hard-ball systems in and out of equilibrium

23. February 2007.

Abstract: The dynamical instability of many-body systems is described by a set of rate constants, the Lyapunov spectrum. Here, we review the computation and some of the properties of such spectra for hard-ball fluids. We demonstrate that the perturbations associated with the large exponents are localized in physical space. However, perturbation vectors connected with the smallest positive Lyapunov exponents exhibit coherent patterns in space, to which we refer as "Lyapunov modes". These are prominent features of hard-ball systems, but rather elusive for particles interacting with a soft potential. Using Fourier-transformation methods, they are shown to exist also in soft-particle systems. We discuss the symmetry properties of the modes and remark on their dynamics. In nonequilibrium stationary states the sum of all Lyapunov exponents is negative, an indication of the fractal nature of the phase-space probability density. We review some recent results for such systems with dynamical or stochastic temperature control.

Michal Rams (Polish Academy of Sciences)

The entropy of Lyapunov-optimizing measures for some matrix cocycles.

11. April 2014.

Abstract: Let $\{A_i\}_{i=1}^k$ be a finite family of matrices from $GL(2,\mathbb{R})$. For a sequence $\omega=\omega_1 \omega_2\ldots \in \{1,\ldots,k\}^{\mathbb{N}}$ consider the maximal Lyapunov exponent of the corresponding product of matrices: $$\lambda(\omega) = \lim_{n\to\infty} \frac 1n \log || A_{\omega_n}\ldots A_{\omega_1}||$$ (wherever it is defined). We want to investigate the sequences for which the Lyapunov exponent takes an extremal (minimal or maximal) value.
This is a natural generalization of standard multifractal questions, with the important difference that the product of matrices is not commutative (hence, the Lyapunov exponent behaves very differently than a usual Birkhoff limit for a real-valued potential).
The result: under some natural assumptions we are able to show that the sets of sequences for which the extremal values of the Lyapunov exponent are achieved are small, in the following sense: all the exponent-minimizing and exponent-maximizing shift-invariant measures have entropy zero. This is a joint work with Jairo Bochi.

Michal Rams (Institute of Mathematics, Polish Academy of Sciences)

Porcupines.

2. December 2011.

Abstract: The pressure function of a transitive (nonuniformly) hyperbolic dynamical system does not need to be smooth. Its singularities correspond to a division of the system into two parts, with different behaviour. Because of transitivity, one of the subsystems must be a 'boundary' one, approached by trajectories of the other subsystem, but because of some mechanism it has a distinct dynamical properties. The typical examples are parabolic maps (like Manneville-Pomeau; the small subsystem is just the parabolic point) and Chebyshev polynomials (like $z\to z^2-2$; the small subsystem is the periodic orbit $\{-2,2\}$, separated from the big subsystem by the critical point at 0). In both cases the smaller subsystem has entropy zero. I'm going to present (the only one known to me) example of a smooth transitive nonuniformly hyperbolic dynamical system with a nondifferentiable pressure function (actually, with the gap in the Lyapunov spectrum) for which both subsystems have positive entropy. It will be then used to construct another example, of a compact invertible system for which the Lyapunov spectrum is not time-reversible. It is a joint work with Lorenzo Diaz (PUC) and Katrin Gelfert (UFRJ).

Michal Rams (Varso)

Contracting on average iterated function systems.

17. November 2008.

Mixing rates of stochastic energy exchange models with degenerate rate functions.

8. February 2013.

Abstract: In recent years, stochastic energy exchange systems of locally confined particles in interaction have been studied intensively, as accessible models for the rigorous study of the derivation of Fourier's law from microscopic dynamics of mechanical origin. As a generalization of them, Grigo et al. introduced a class of pure jump Markov processes of energies and studied the spectral gap of them under the assumption that the rate function of the energy exchange is uniformly positive. In this talk, I will consider the case where the rate function does not have a uniform lower bound, and give a lower bound estimate of the spectral gap. The hydrodynamic behavior of these systems will be also discussed.

Scheuring István (ELTE Növényrendszertani és Ökológia Tanszék, Ökológiai és Elméleti Biológiai Kutatócsoport)

Stratégiák fixációja evolúciós mátrixjátékokban véges populáció-méret esetén

7. April 2006.

Abstract: 2 x 2-es mátrixjátékok sztochasztikus evolúciós dinamikáját tanulmányoztuk véges populációkban. Az egyes stratégiák fixációs tulajdonságait vizsgáltuk: Milyen valószínûséggel fixálódik egy mutáns stratégia és mennyi a fixáció idejéne várható értéke? Megmutattuk, hogy tetszõleges 2 x 2-es mátrixjátékok esetén mindkét stratégia azonos fixációs idõvel rendelkezik. A fixációs valószínûségeket nagy populáció-méret határesetben tudtuk kiszámolni. Megmutattuk, hogy a fixáció "gyors" ha a végtelen populáció-méret határesetben van legalább egy evolúciósan stabil stratégia (ESS), míg a fixáció "lassú" ha a végtelen határesetben nincs ESS.

Klaus Schmidt (University of Vienna and Erwin Schrödinger Institute)

Entropy and periodic points for (commuting) group automorphisms.

1. March 2013.

Abstract: I am going to discuss the connection between entropy and the logarithmic growth rate of the number of periodic points for Z- and Z^d-actions by automorphisms of compact abelian groups.

Klaus Schmidt (University of Vienna and Erwin Schrödinger Institute)

Ergodic Theory and Number Theory -- On the work of Elon Lindenstrauss.

11. November 2011.

Abstract: In this lecture I will describe some of the work by Elon Lindenstrauss and his collaborators for which he was awarded the Fields Medal last year.

Klaus Schmidt (University of Vienna and Erwin Schrödinger Institute)

On some of the differences between Z and Z^2 in dynamics.

10. November 2011.

Abstract: The aim of this lecture is a gentle introduction to algebraic $Z^d$-actions with $d\ge2$, and to the problems and phenomena arising in the transition from $d=1$ to $d>1$. This lecture will prepare the ground for the Friday lecture on Elon Lindenstrauss' work.

Klaus Schmidt (ESI and Universität Wien)

Sandpiles and the Harmonic Model

27. February 2009.

Pablo Shmerkin (University of Surrey)

21. September 2012.

Abstract: The Hausdorff dimension of sets invariant under conformal dynamical systems can often be realized as the zero of certain natural pressure equation (going back to Bowen). This pressure is usually continuous as a function of the defining dynamics, in the appropriate topology, and hence so is the Hausdorff dimension of the invariant set. The situation is dramatically more complicated in the non-conformal situation where, nevertheless, a subadditive version of the pressure equation, involving singular values of a matrix cocycle, is crucial. A natural question is therefore whether this subadditive pressure is also a continuous function of the dynamics (or, what is the same, of the associated cocycle). We resolve this in the affirmative in many important situations, in particular answering a question of Falconer and Sloan. This is joint work with De-Jun Feng (Chinese University of Hong Kong).

Pablo Shmerkin (University of Manchester)

Geometric rigidity of $\times \beta$-invariant measures.

27. May 2011.

Abstract: The celebrated theorem of Rudolph-Johnson says that if $p,q\ge 2$ are two integers which are not common powers of a same integer, then any two measures on the circle which are invariant and ergodic under $\times p, \times q$ respectively are singular unless they have zero entropy or are Lebesgue. We generalize this to the case in which $p>1$ is a Pisot number and $q>1$ is arbitrary. Other geometric improvements on Rudolph-Johnson's theorem will be discussed. This is joint work with M. Hochman.

Nándor Simányi (University of Alabama at Birmingham)

The Boltzmann-Sinai Hypothesis: a Mystery Solved.

8. May 2014.

Abstract: The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai's modern formulation of Ludwig Boltzmann's statistical hypothesis in physics, actually as a conjecture: Every hard ball system on a flat torus is (completely hyperbolic and) ergodic (i. e. "chaotic", by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities. In the half century since its inception quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Quite recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality. In the talk I plan to sketch some important technical details of the concluding part of the proof.

Simányi Nándor (UAB)

Rugalmas korongrendszerek szélcsatornában.

13. May 2011.

Abstract: Két párhuzamos egyenes fal között mozgó kemény golyók olyan kétdimenziós rendszereinek az ergodikus tulajdonságait vizsgáljuk, amik a falak irányában periodikusak, a részecskék közötti kölcsönhatás rugalmas ütközés, míg a falakról való visszaverõdés megváltoztatja a sebességek irányát (de nem a hosszát) a rugalmas ütközéshez képest. Mindezt olyanképpen, hogy az egyik fallal való ütközés irányváltoztatása egy jobbra mutató driftet jelent, míg a másik fallal való ütközések eredményezte drift az elõzõvel egyenlõ, de azzal ellentétes irányú (úgynevezett shear flow). Az elõadásban megfogalmazok számítógépes szimulációval indukált, illetve alátámasztott sejtéseket, fenomenológiai szintû (fizikai) és heurisztikus érveléseket és néhány szigorú bizonyítást vázolok. Az eddigi eredmények még csak kezdetiek és Nikolai Chernovval és Alexey Korepanovval közösek.

Simányi Nándor (Birmingham, Alabama)

Az Alaptétel 2D biliárdokra Ansatz nélkül.

21. May 2009.

Abstract: A biliárdokra vonatkozó lokális ergodicitási tétel (úgyis, mint az "Alaptétel") elégséges feltételt ad arra, hogy egy fázispont valamely környezete egyetlen ergodikus komponenshez tartozzék (modulo a nulla mértékû halmazok természetesen). Ez a tétel kulcsfontosságú matematikai biliárdok és a hozzájuk hasonló, szinguláris, nem egyenletesen hiperbolikus dinamikai rendszerek ergodikussága bizonyításában. Sajnos azonban az Alaptétel bizonyítása használ egy finom globális feltételt, az ún. Csernov-Szinaj Ansatz-ot, amit eléggé körülményes ellenõrizni fizikailag releváns rendszerekre, többek között a kemény golyók gázára. Az elõadás témája egy N. Csernovval tavaly elért közös eredmény, kétdimenziós biliárdok lokális ergodikussági tételének egy új bizonyítása, kikerülve az Ansatz felhasználását.

Simányi Nándor (UAB)

Invariáns hiperfelületek félig-szóró biliárdokban

19. July 2006.

Abstract: This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of $J$ grows at least linearly in time. Our proof is independent of the solution of the Boltzmann -Sinai hypothesis, and we provide a complete and self-contained argument here. (Joint work with N.I.Chernov)

Simon Károly (BME MI)

The intersection of the Sierpinski Carpet with straight lines.

19. March 2010.

Abstract: One of the most popular self-similar fractal set is the Sierpinski carpet. To obtain it, we partition the unit square into 9 congruent copies and throw away the one in the middle. We repeat the same process for the remaining squares ad infinitum. The set we obtained is the Sierpinski carpet. The intersection of the Sierpinski carpet with a straight line is a fractal set itself. It is known that the size (Hausdorff dimension) of the Sierpinski carpet and a straight line is different for different lines. However, for many (in some natural sense) lines the Hausdorff dimension of this intersection is equal to the Hausdorff dimension of the carpet (which is log 8/log 3) minus one. In this talk I will speak about our joint result with Anthony Manning concerning the exceptional behavior of the intersection of the Sierpinski carpet with lines of rational slopes. (Joint work with A. Manning, University of Warwick, UK)

Boris Solomyak (University of Washington, Seattle)

Multifractal structure of Bernoulli convolutions.

28. May 2010.

Abstract: Infinite Bernoulli convolutions is perhaps the simplest and most studied family of overlapping self-similar measures. It is known that for almost all parameters $\lambda$ in (0.5,1) the symmetric Bernoulli convolution is absolutely continuous and equivalent to Lebesgue measure on its support. However, even then the measure may be `multifractal'' and the density may have many zeros in the interior of its support. This phenomenon will be explored in the talk, which is based on a joint work with Thomas Jordan and Pablo Shmerkin.

Mikko Stenlund (University of Helsinki)

An Adiabatic Dynamical System as a Stochastic Process.

25. October 2013.

Abstract: The statistical properties of dynamical systems are traditionally studied in the context of their invariant measures. Motivated by non-equilibrium phenomena in nature, we wish to step out of the above setup. To this end, we introduce a model whose characteristics change slowly with time; hence the word adiabatic in the title. Solving a martingale problem in the spirit of Stroock and Varadhan, we show that repeated observations of the state of the system yield a certain stochastic diffusion process.

Sütõ András (MTA SZFKI)

Korrelációs egyenlõtlenség merev gömbök rendszeréhez.

15. February 2013.

Absztrakt: A d-dimenziós euklideszi térben minden lehetséges módon el kívánunk helyezni n egyforma merev gömböt úgy, hogy a középpontjaik egy elõre megadott korlátos, Lebesgue-mérhetõ tartományba essenek. A sejtés az, hogy az n+1-edik gömb számára kizárt térfogat átlaga n-nel nõ. Az állítás természetes de nem triviális, mert az átlagoló mérték maga is függ n-tõl. Az elõadásban ismertetek néhány részeredményt és egy lehetséges bizonyítási utat.

Domokos Szász (BME MI)

On the work of Nikolai Chernov I.

19. September 2014.

Abstract: On August 7, 2014, the great mathematician, Nikolai Chernov, our close friend passed away. In this seminar we commemorate him by explaining some of his most exceptional ideas.

Domokos Szász (BME MI)

Spectral gap in a stochastic energy model.

7. October 2011.

Abstract: In 2008, Pierre Gaspard and Thomas Gilbert formulated a two step strategy for obtaining the heat equation from microscopic principles through a Markov jump process of energies. Their mechanical model was one of localized hard disks (balls). Here we consider a linear chain of N particles each carrying an energy. The evolution of the energies is governed by a continuous time, pure jump Markov process. The interactions are nearest neighbor ones preserving the energy. (This family of stochastic evolutions contains the models of Gaspard and Gilbert.) A lower bound (in terms of N) is presented for the spectral gap of the Markov generator under the assumption that the stationary distributions are reversible and, moreover, reversible states are also characterized. Our spectral bound is just the necessary one for possibly deriving the heat equation. Joint work with A. Grigo and K. Khanin.

Dalia Terhesiu (University of Vienna)

Sharp mixing rates for some finite and infinite measure preserving semiflows.

31. January 2014.

Abstract: In recent joint work with Ian Melbourne we obtain sharp mixing rates for Gibbs Markov semiflows preserving an infinite measure. In work in progress also with Melbourne, we obtain lower and upper polynomial bounds for the correlation decay in the setting of Gibbs Markov semiflow. In this talk I will describe the general framework/method of proof and some of the technicalities involved.

Tamás Tél (Institute for Theoretical Physics Eotvos University)

Absorbing billiards and other dynamical systems.

8. May 2014.

Abstract: Motivated by applications in optics and acoustics we develop a dynamical-system approach to describe absorption in chaotic systems. We introduce an operator formalism from which we obtain (i) a general formula for the escape rate in terms of the natural conditionally invariant measure of the system, (ii) an increased multifractality when compared to the spectrum of dimensions Dq obtained without taking absorption and return times into account. Simulations in the cardioid billiard confirm these results.

Joerg Thuswaldner(Leoben)

Topology of self similar fractals

23. November 2007.

Abstract: The lecture aims at surveying some results on topological properties of self similar sets. We give results on their connectivity, on cut points, on homeomorphy to an arc or to a closed disk as well as on their fundamental group.

Mike Todd (St Andrews)

Dynamical systems with holes: slow mixing cases.

27. February 2015.

Abstract: Fernandez and Demers studied the statistical properties of the Manneville-Pomeau map with the physical measure when a hole is put in the system, overcoming some of the problems caused by subexponential mixing. I will discuss the same setup, but with a class of natural equilibrium states. We find conditionally invariant measures and give precise information on the transitions between the fast exponentially mixing, the slow exponentially mixing and the subexponentially mixing phases. This is joint work with Mark Demers.

Tóth Bálint (BME & University of Bristol)

Superdiffusive CLT for the periodic Lorentz gas in the Boltzmann-Grad limit.

4. April 2014.

Abstract: We prove central limit theorem (CLT) for the displacement of a test particle in periodic Lorentz gas in the limit of low scatterer density (Boltzmann-Grad limit), under $\sqrt{t \log t}$ scaling. The result holds in any dimension and for a general class of finite-range scattering potential. It complements earlier results of P. Bleher (1992), respectively, D. Szász and T. Varjú (2007), where a similar limit theorem is conjectured, respectively, proved for the 2d periodic Lorentz gas with infinite horizon. The talk is based on joint work with Jens Marklof (Bristol).

Imre Péter Tóth (BME MI)

Complexity of singularities and growth of unstable manifolds in planar dispersing billiards with corner points.

14. December 2012.

Abstract: Exponential decay of correlations and many other strong statistical properties were proven for several classes of planar dispersing billiard maps, starting from the work of Young in 1998 and Chernov in 1999. The proofs, which mostly use the Young tower technique, rely on understanding the growth of unstable manifolds in the presence of singularities. However, in the case when the billiard table has corner points, the proofs remain conditional, assuming the so-called "subexponential complexity" property of the singularity structure. This property means that as we iterate the map, the number of smooth components meeting at any point grows at most subexponentially. We now fill this gap by a detailed study of the evolution of these smooth components. One needs to take into account the "branching" of the billiard map at phase points that reach corners, which originates from non-continuity of the flow. In the finite horizon case we are actually able to check the subexponential complexity property, while in the infinite horizon case it's better to avoid using complexity and prove the "growth lemma" of unstable manifolds directly. This work is joint with Jacopo de Simoi.

Tóth Imre Péter (BME MI)

Separation of time scales and averaging in hyperbolic dynamical systems.

9. March 2012.

Abstract: If we want to understand how the chaotic, hyperbolic behaviour of deterministic Physical models leads to phenomena observed in Statistical Physics, a useful and interesting approach is to approximate the system with a stochastic process. If the observables of the system that we wish to study change "slowly" compared to some "fast evolving" coordinates, and the behaviour of these "fast variables" is suitably chaotic, then, while the slow variables are nearly constant, the fast have enough time to "reach equilibrium", so the slow variables will be "driven" by some "average values", or possibli equilibrium fluctuations of the fast ones. This phenomenon can be phrased in a mathematically rigorous way, if the system has a parameter, which can be tuned to approach a limiting case when the slow variables become infinitely slow. In sich a limit, after rescaling time properly, the process of slow variables may converge to some "process living on its own". Depending on whether the "average force" driving the slow variables is zero or not, one may have to use different scalings to obtain a deterministic or a Markov limit process.
Mathematical study of this area is a currently hot topic. In this talk I discuss one of the fundamental works of the topic, a paper of Dmitry Dolgopyat from 2005. In this paper he proves different cases of the above limit behaviour under technical conditions on "strong mixing" of the fast variables. These contitions are phrased using one of the most modern tools in the science of hyperbolic dynamical systems, the "standard pairs".

Tóth Imre Péter (BME MI)

About the complexity of the singularity set in planar dispersing billiards: the complexity is typically finite.

25. November 2011.

Abstract: In the study of statistical properties of hyperbolic dynamical systems with singularities, understanding the singularity structure plays a decisive role. Under complexity of the singularities we mean the function, which tells that if we run the dynamics for time t, what is the maximum number of component into which a small neighbourhood of a phase point can break up. Estimating this function is necessary to prove one of the most important statistical properties, exponential decay of correlations (with the existing techniques). In dispersing billiards this is a serious problem, because -- while in the simplest planar case the complexity is known to grow at most linearly -- in dimensions greater than two, no usable estimate is known. Although it is natural to conjecture that in a _typical_ configuration the complexity is finite, because every phase point can be singular at most a finite number of times, during its entire trajectory. In the talk I prove this conjecture for the simplest possible case: planar dispersing billiards with finite horizon, with typicality meant in the C^3 topological sense (as holding on a residual set). Even this is not easy: one has to fight the problem of "recollisions", that the effect on the trajectory of a change in the configuration is hard to follow, because the trajectory, during its history, can return several times to the configuration point effected by the change. In the proof I present, the way out is to study the effect of perturbations in a dynamical way, making use of the hyperbolicity of the system.

Tóth Imre Péter (BME MI)

Hõvezetési egyenlet energiát megõrzõ dinamikák csatolt rendszerére.

27. May 2011.

Abstract: Az elõadásban Jean Bricmont és Antti Kupiainen eredményét ismertetem, melyben egy determinisztikus hõvezetés-modellre hajtják végre a hidrodinamikai határátmenetet (diffúzív skálázással), és vezetik le a hõvezetési egyenletet matematikailag szigorúan. A modellben a d-dimenziós egészrács minden pontjában egy egyszerû dinamikai rendszer ül, aminek fázistere egy "gyors" és egy "lassú" változóból áll. A csatolatlan dinamika a lassú változót (ami az energia szerepét játsza) megõrzi, a gyors változóban pedig gyorsan kever. Ehhez járul még egy gyenge csatolás, ami a szomszédos rácspontok közötti energia-cseréért felel, de az összenergiát megõrzi. Erõs regularitási feltételek mellett a szerzõk ennek a rendszernek a diffúzív határviselkedését tudják bizonyítani, éspedig nem csak a "gyenge csatolás" határesetben, hanem a csatolási paraméter kicsi, de pozitív értékeire is. A bizonyítás eszköze egy renorm-csoport-transzformáció, kulcs-észrevétele pedig a modellnek a véletlen közegben való bolyongással mutatott analógiája. A tárgyalt cikk elérhetõ itt

Tóth Imre Péter (BME MI)

Aktuális mechanikai és részben mechanikai hõvezetés-modellek.

20. May 2011.

Abstract: Az elõadás célja az elmúlt pár év néhány eredményének áttekintése a hõvezetés mechanikai modelljeinek matematikai vizsgálata terén. Ehhez az áprilisban Torntóban rendezett "Workshop on the Fourier Law and Related Topics" konferencián ismertetett munkákból tárgyalok néhányat, amik kifejezetten a Fourier-törvény érvényességét vizsgálják sokrészecskés kölcsönható modellekben. A modellek egy része tisztán determinisztikus, más részében a determinisztikus mechanikai rendszer sztochasztikus zajjal van kiegészítve. A tárgyalt cikkek Dmitry Dolgopyat, Antti Kupiainen, Carlangelo Liverani, Stefano Olla és szerzõtársaik eredményei.

Tóth Imre Péter (BME MI)

Félig szóró biliárdok alaptétele Ansatz nélkül, 2 dimenzióban.

12. November 2010.

Abstract: A biliárd-elmélet egyik legfontosabb és legismertebb nyitott kérdése a Boltzmann-Sinai ergodikus sejtés, mely szerint egy legalább két kemény golyó alkotta biliárd a tóruszon mindig ergodikus - akárhány dimenzióban (ha a sugarak elég kicsik, hogy a fázistér összefüggõ legyen). A sejtés bizonyításában az elmúlt 5-10 évben rengeteg elõrelépés történt, elsõsorban Simányi Nándor munkái nyomán. Mára az egyetlen hiányzó láncszem, hogy a lokális ergodicitás bizonyításában fel kell tenni az úgynevezett "Chernov-Sinai Ansatz"-ot. Ez egy látszólag technikai feltevés, amit azonban máig nem sikerült bizonyítani. Ebben az elõadásban a Boltzmann-Sinai sejtés bizonyítása irányában tett legújabb lépést ismertetem. Ez Simányi Nándor és Nikolai Chernov cikke, amiben azt mutatják meg, hogy hogyan lehet a lokális ergodicitást az Ansatz nélkül bizonyítani. Az eredmény gyengéje, hogy csak 2 dimenzióban érvényes, így a sejtés általános esetben továbbra is nyitott. Az elõadásban elmondom a bizonyítás alapját képezõ új felismerést, és beszélek arról, hogy hogyan lehet szerintem a módszert magasabb dimenzióban is - a Boltzmann-Sinai sejtés általános bizonyítására - alkalmazni.

Tóth Imre Péter (BME MI)

Hõvezetési modell gyengén kölcsönható lokalizált biliárd korongokkal

1. October 2010.

Abstract: Olyan hõvezetési modellt vizsgálok, ahol rácspontok közelében lokalizált biliárd korongok hatnak kölcsön konzervatív erõk révén. Ez a népszerû Gaspard-Gilbert modell egy természetes módosítása. A gyenge csatolás határesetben (az idõ megfelelõ skálázása mellett) az egyes rácspontokon tárolt energiák egy Markov-lánc szerint változnak. Más szóval ebben a határesetben az energiák folyamata egy kölcsönható részecskerendszer, ahol azonban az energia értékei, sõt a trajektóriák is, folytonosak. Ennek a kölcsönható részecskerendszernek a hidrodinamikai határviselkedését elég pontosan sikerült megérteni részben szigorú, részben heurisztikus és numerikus eszközökkel. Az jön ki, hogy a hõvezetési együttható hõmérséklet-függése $T^{-3/2}$, ami meglepõen jól egyezik bizonyos kísérleti adatokkal. Ez azt is jelenti, hogy - bár a Gaspard-Gilbert modell módosításának motivációja a matematikai tárgyalás könnyebbé tétele volt - ez a módosított modell bizonyos szempontból jobb, reálisabb, mint az eredeti. Az elõadás során elsõsorban a kölcsönható részecskerendszer jellemzésére fogok összpontosítani. Elmondom, hogy ez miért egy diffúziós folyamat, és hogyan jelenik meg a gyenge csatolás határesetben. Külön hangsúlyozom a rendszer szimmetriáit, azon belül is a skálázási tulajdonságait, amik a hidrodinamika megértésében kulcsszerepet játszanak.

Tóth Imre Péter

Korrelációlecsengés tornyok nélkül

19. January 2007.

Tóth Imre Péter

Ergodicity and correlation decay in billiards (PhD dolgozat nyilvános védése)

28. April 2006.

Tóth Imre Péter (BME MI)

Exponenciális korrelációlecsengés magas dimenziós szóró biliárdokban (folytatás)

3. March 2006.

Tóth Imre Péter (BME MI)

Exponenciális korrelációlecsengés magas dimenziós szóró biliárdokban

24. February 2006.

Varjú Tamás (BME MI)

Horizons in multidimensional billiards.

8. March 2012.

Abstract: Zacherl et. al (1986) and Bleher (1992) conjectured and gave heuristic arguments for the (weak) superdiffusivity of 2D Lorentz processes with infinite horizon. In 2007, we could give a rigorous proof with D. Szász and could also obtain an explicit form of the superdiffusivity coefficient in terms of geometric parameters of the billiard involved. In 2011, C. Dettmann described heuristically the much richer horizon structure in the multidimensional case and formulated conjectures that would lead to a generalization of the 'simple' 2D form of the superdiffusivity coefficient. Here we establish Dettmann's Conjecture 1 by simple geometric probabilistic arguments. The results are joint with P. Nándori and D. Szász.

Varjú Tamás (BME MI)

Síkbeli Lorentz folyamat bolyongási tulajdonságai

6. March 2008.

Abstract: Darling-Kac, illetve Erdõs-Taylor bolyongásra bizonyított eredményeit általánosítjuk síkbeli, véges horizontú Lorentz-folyamatra. A vizsgált mennyiségek az elsõ visszatérés ideje, az elsõ elérés ideje, lokális idõ, elsõ találkozás ideje, valamint határeloszlás a fázisra ezen nevezetes idõpontokban. Dmitry Dolgopyattal és Szász Domokossal közös eredmények.

Varjú Tamás (BME MI)

Biliárd önhasonló asztalon

28. September 2007.

Abstract: "A biliárd konstans és kicsi külsõ tér mellett egy érdekes modell. Tudjuk, hogy invariáns mértéke szinguláris, hogy a részecske sebessége pozitív várható értékû a külsõ tér irányában és még sok mindent. Ezt a tudásunkat átmenthetjük egy speciális önhasonló biliárdasztal külsõ tér nélküli biliárdjára a Wojtkowski transzformáció segítségével. A két modell hasonlóságairól és különbségeirõl, ezek fizikai vonatkozásáról és néhány matematikai eredményrõl számolok be Gilbert, Barra, Chernov és Dolgopyat cikkei alapján."

Lai-Sang Young (Courant Institute, NYU)

Dispersing billiards with moving scatterers.

15. June 2012.

Abstract: We propose a model of Sinai billiards with moving scatterers, in which the locations of the scatterers may be shifted by small amounts between collisions. Our main result is the exponential loss of memory of initial data, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems. (Joint work with Mikko Stenlund and Hongkun Zhang)

Lai-Sang Young (Courant Institute, NYU)

Measuring dynamical complexity.

13. June 2012.

Abstract: I will discuss three ways to capture dynamical complexity: (A) hyperbolicity, a geometric characterization of instability, (B) entropy, an information-theoretical approach to capturing the randomness of dynamical events, and (C) correlation decay or memory loss as a function of time. I will review these ideas in nontechnical terms, discuss how they are related, and give a very brief (and somewhat personal) survey of the progress made in the last decades. To illustrate these ideas, I will use a concrete example, namely that of shear-induced chaos in periodically kicked oscillators. The idea of this example goes back to van der Pol nearly 100 years ago, but the mathematics was done only recently.

Roland Zweimüller (University of Vienna)

Return- and Hitting-time limits for rare events in infinite measure preserving systems.

19. September 2014.

Abstract: While there exists a large body of work analysing "return- and hitting-time statistics" (limit laws for the return- or hitting times of small sets) for finite measure preserving dynamical systems, very little is known in the infinite measure (null recurrent) case. I will report on joint work with F Pene and B Saussol (Brest, F) which clarifies what to expect in situations with reasonable ergodic/mixing properties. Specific results for dynamical systems concern cylinders of Markov maps. Our considerations also yield results for null-recurrent Markov chains which (somewhat surprisingly) seem to be new.

Roland Zweimüller (University of Vienna)

Dissipative dynamics as a tool for conservative systems.

14. December 2012.

Abstract: I will report on joint work with David Kocheim in which we develop new conditional limit theorems for Gibbs-Markov systems with holes and use these to get a better understanding of stochastic properties of certain conservative infinite measure preserving transformations.