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Motivation (excerpts from the proposal accepted in
January 2006)
In many ways, the present
semester at ESI is a continuation of a similar effort at ESI in 1996.
Then, the main hope was that new progress could be made through the
discovery of common threads that run through the analyses of quite
different systems such as the Henon map or billiards. In some sense,
Lai-Sang Young's 1998 Annals of Math. paper [Y] represents an
embodiment of this program.
Since then, techniques from different topics and directions have
merged, leading to radically new developments in non-uniformly
hyperbolic and partially hyperbolic systems, in hyperbolic systems with
singularities, in "large" dynamical systems such as coupled map
lattices, and in the application of dynamical systems ideas to
nonequilibrium statistical mechanics. There have been spectacular
results in the field, where, in particular, we refer to Refs. [D],
[Sim2], [T], and to Refs. [WY], [ChD].
We believe that it is time to consolidate these
gains and to explore future directions. The new techniques involve a
variety of novel ideas and are so elaborate that only a joint effort of
the leading experts seems to be able to master them. The trimester at
ESI is expected to provide a platform for such a collaborative effort.
Much of the success of the 1996 semester was
derived from its interdisciplinary character. Approximately one third
of the participants were physicists. We wish to continue in this
tradition and intend to bring together researchers in the theory of
hyperbolic dynamics and experts in computational dynamics and
statistical physics.
Problems to focus on in the ESI
semester:
Here, we list some directions in which we
expect that progress will be made.
1. Multidimensional Sinai billiards:
Chernov generalized Young's ideas on Markov extensions [Y] to
multidimensional Anosov systems with singularities. However, this
generalization does not cover Sinai billiards where, close to the
singularities, the derivative becomes unbounded. As observed by
Bálint, Chernov, Szász, and Tóth [BChSzT], this
unboundedness leads to the nondifferentiability of the images of the
singularities in dimension d ≥3, circumstances that have so far blocked
the generalization of Young's construction and, consequently, the
control of correlation decay for multidimensional Sinai billiards.
Since all other ingredients of hyperbolic theory needed for this task
are under control, this is the only remaining obstacle.
2. Semi-dispersing billiards:
Success in the previous problem may be followed by results on the
control of correlation decay for semi-dispersing billiards. On the
level of local ergodicity, a striking problem is to generalize the
local ergodicity theorem to the smooth case by dropping the unpleasant
algebraicity condition.
An encouraging step forward is the extension of the local ergodicity
theorem to certain smooth dispersing billiards by [Bat], [BF] .
3. Ergodicity of hard balls:
Simányi's proof of the ergodicity for typical hard ball systems
in arbitrary dimensions needs to be simplified in order to obtain a
proof of ergodicity of hard ball systems in full generality. Then,
there is the problem of balls in a box.
4. Hyperbolic flows with singularities:
Improvements of Dolgopyat's methods for the stochastic properties of
hyperbolic flows with singularities (including billiard flows) seem now
to be within reach.
5. Billiards defined by potentials:
It is known that ergodic potentials are exceptional. It is a challenge
to find such potentials in higher dimensions, generalizing the
2-dimensional results of Donnay-Liverani. The recent works of
Bálint and Tóth aim in that direction.
6. Mechanical models with rotating disks:
A new class of mechanical models involving the interaction of particles
with rotating disks was introduced [LLM], which involves the exchange
of energy between translational and rotational degrees of freedom. From
the technical point of view, depending on the ratio of the masses of
the disks and particles, these models may have varying degrees of
hyperbolicity, but they are not uniformly hyperbolic. Deciphering their
statistical properties poses new challenges for billiard techniques.
7. Mechanical models of Brownian motion: It is
evident that the ideas and methods of Ref. [ChD] provide fundamental
new insights. By their deeper and - hopefully - simplified
understanding, it is reasonable to expect that the methods can be
extended beyond the limitations of their present conditions, and to
other instructive models as well.
8. Rank one attractors: These
ideas grew out of earlier work on Hénon maps. By framing the
ideas in terms of rank-one maps (i.e. maps with a single direction of
instability and strong contraction in all other directions), Wang and
Young identified a tractable class of strange attractors that arise
naturally. It remains to develop a fuller dynamical picture of these
maps, including their geometric and statistical properties, and to
establish their presence in various natural physical and mechanical
systems defined by ODEs and PDEs.
9. Coupled map lattices: While
the situation involving weakly coupled hyperbolic maps is now
relatively clear, much remains to be understood in situations where the
coupling strength is stronger (but not strong enough for
synchronization to occur), and in situations where the constituent maps
are not uniformly hyperbolic.
10. Nonequilibrium dynamics: A
class of problems related to those above has to do with generalized
conduction or transport. As an example we mention chains of coupled
particles, where the ends of the chain are connected to heat baths
(thermostats) leading to a flow of particles and/or energy along the
chain.
Among the questions of interest are the existence of (linear)
temperature profiles and descriptions of the particle density along the
chain. In Ref. [EY], models similar to those in Ref. [LLM] were studied
(see item 6). Stochastic and time-reversible boundaries were used.
Problems abound in this area. Such systems play a crucial role in the
current revival of nonequilibrium statistical mechanics [RB].
11. Comupter-assisted studies: In
addition to the use of computer explorations to give insight into those
parts of nonlinear dynamics not (yet) accessible to analytic studies,
computer assisted methods can be used to resolve many questions in
dynamical systems on a more rigorous level. For example, the presence
of Shilnikov-type homoclinic orbits may be verified and the statistical
properties of the attractors near them investigated.
12. Theoretical justifications of
computations in dynamics: There is an urgent need to better
understand the theoretical background and meaning of numerical
simulations (see e.g. the Refs. [K], [DSz], and [FL] )
13. Infinite hyperbolic dynamics:
There is strong physical motivation for understanding infinite
measure hyperbolic dynamical systems, in particular, the Lorentz
process. For recent progress in this
direction see, for example, [SzV] and related works.
Bibliography:
[BChSzT] P. Bálint and N. Chernov and
D. Szász and I. P. Tóth, Geometry of
multidimensional dispersing billiards, Astérisque, 286,
119-150 (2003).
[Bat] P. Batchourine, On the structure of the singularity
manifolds of dispersing billiards, arXiv:math/0505620
[BF] P. Batchourine and Ch. Fefferman, The
volume near the zeroes of a smooth function, preprint.
[ChD] N. Chernov and D. Dolgopyat,
Brownian motion I. manuscript, pp. 192, 2005.
[D] D. Dolgopyat, On decay of correlations in Anosov flows,
Ann. of Math., 147, 357-390 (1998).
[DSz] G. Domokos and D. Szász, Ulam's
Scheme Revisited: Digital Modeling of Chaotic Attractors via
Micro-Perturbations, Discrete and Continuous Dyn. Systems, Ser. A.
9, 859-876 (2003).
[EY] J.-P. Eckmann and L.-S. Young, Temperature
profiles in Hamiltonian heat conduction, Europhys. Lett. 68,790-796
(2004).
[FL] P. P. Flockermann, Discretizations
of expanding maps, PhD thesis, Dept. of Math., ETH-Zurich] , 2001;
and a paper in preparation, with the same title, by P. P. Flockermann
and O. E. Lanford
[K] Yu. Kifer, Computations in dynamical
systems via random pertubations.
Discrete and Continuous Dyn. Systems, 3, 457-476
(1997).
[LLM] H. Larralde, F. Leyvraz, C.
Mejia-Monasterio, Transport properties of a modified Lorentz gas,
J. Stat. Phys. 113, 197 (2003).
[RB] L. Rey-Bellet, Statistical mechanics
of anharmonic lattices, Contemporary Matheamtics ,
arXiv:math/0303021.
[Sim2] N. Simányi, The
Boltzmann-Sinai ergodic hypothesis in full generality
(without exceptional models), arXiv:math/0510022.
[SzV] D. Szász , T Varjú, Local limit theorem for the Lorentz
process and its recurrence on the plane, Ergodic
Theory and Dynamical Systems (2004), 24, 257-278
[T] W. Tucker, A Rigorous ODE Solver and
Smale's 14th Problem,
Found. Comput. Math., 2, 53-117 (2002).
[WY] Q. Wang, L.-S. Young, Toward a theory
of rank-one attractors, to appear in Annals of Math.
[Y] L.-S. Young, Statistical properties of
systems with some hyperbolicity
including certain biliards, Ann. of Math., 147,
585-650 (1998).
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