Schedule for: 18w5061 - New Developments in Open Dynamical Systems and Their Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will consider a class of deterministic walks in random environment and discuss a strategy for studying their long time behavior. I will describe some simple examples and discuss the possibility of applying this strategy to the random Lorenz gas (work in collaboration with Romain Aimino)

We consider a model of globally coupled circle maps, the finite version of which was studied in the works of Koiller-Young, Fernandez and Balint-Selley. In the continuum version the state of the system is described by a density on the circle. For a fairly general class of expanding circle maps we show that, for sufficiently small coupling, there is a unique invariant density. For sufficiently strong coupling the density converges to a Dirac mass that moves chaotically on the circle. This is joint work with G. Keller, F. Selley and I.P. Toth.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk we discuss recent developments concerning stability properties of non-autonomous dynamical systems, motivated by? the ergodic theoretical study of random, forced or time-dependent systems and their coherent structures. In the setting of random interval maps we present results about stability of random absolutely continuous invariant measures. In the context of multiplicative ergodic theory, we discuss results on stability of Lyapunov exponents and Oseledets spaces in nite and innite-dimensional settings. This is based on joint works with Gary Froyland, Rua Murray and Anthony Quas.

Optional: Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

M. Kac popularized the question {\em Can you hear the shape of a drum?} Mathematically, consider a bounded planar domain $\Omega$ and the associated Dirichlet problem $\Delta u + \lambda^2 u = 0$ with $u|_{\partial \Omega}$ = 0. The set of $\lambda$s such that this equation has a solution, denoted $\mathcal{L}(\Omega)$ is called the Laplace spectrum of $\Omega.$ Does Laplace spectrum determine $\Omega$? In general, the answer is negative. Consider the billiard problem inside ?. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any generic axis symmetric planar domain with is dynamically spectrally rigid, i.e. can't be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. This is joint works with J. De Simoi, A. Figalli, and J. De Simoi, Q. Wei.

In 1949, Fermi proposed a mechanism for the heating of particles in cosmic rays. He suggested that on average, charged particles gain energy from collisions with moving magnetic mirrors since they hit the mirrors more frequently with heads on collisions. Fermi, Ulam and their followers modeled this problem by studying the energy gain of particles moving in billiards with slowly moving boundaries. Until 2010 several examples of such oscillating billiards leading to power- law growth of the particles averaged energy were studied. In 2010 we constructed an oscillating billiard which produces exponential in time growth of the particles energy [1]. The novel mechanism which leads to such an exponential growth is robust and may be extended to arbitrary dimension. Moreover, the exponential rate of the energy gain may be predicted by utilizing adiabatic theory and probabilistic models [2,3]. The extension of these results to billiards with mixed phase space leads to the development of adiabatic theory for non-ergodic systems [4]. Finally, such accelerators lead to a faster energy gain in open systems, when particles are allowed to enter and exit them through a small hole [5]. The implications of this mechanism on transport in extended systems [6] and on equilibration of energy in closed systems like springy billiards will be discussed [7]. These are joint works, mainly with with K. Shah, V. Gelfreich and D. Turaev [1-5],[7] and [6] is with M. Pinkovezky and T. Gilbert: [1] K. Shah, D. Turaev and V. Rom-Kedar, Exponential energy growth in a Fermi accelerator, Phys. Rev. E 81, 056205, 2010. [2] V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev, Robust expo- nential accelerators, PRL 106, 074101, 2011.

I will describe the smooth non-uniformly hyperbolic map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov automorphism near the origin and it is a local (but not small) perturbation. The Katok map was the rst example of an area preserv- ing dieomorphism with non-zero Lyapunov exponents and can be used to construct such dieomorphisms on any surface. I will then discuss the thermodynamical formalism for the Katok map, i.e., demonstrate existence and uniqueness of equilibrium measures associated with the geometric potential and their ergodic properties including decay of cor- relations and the Central Limit Theorem. This is based on recent works with S. Senti and K. Zhang.

I will talk about extending my earlier work on the vanishing noise limit of diusions in noisy heteroclinic networks to longer time scales. In this eld, the results are based on sequential analysis of exit loca- tions and exit times for neighborhoods of unstable equilibria. The new results on exit times and the emergent hierarchical structure are joint with Zsolt Pajor-Gyulai.

In dynamical systems with divided phase space, the vicinity of the boundary between regular and chaotic regions is often "sticky," that is, trapping orbits from the chaotic region for long times. Here, we investigate the stickiness in the simplest mushroom billiard, which has a smooth such boundary, but surprisingly subtle behaviour. As a measure of stickiness, we investigate P(t), the probability of remaining in the mushroom cap for at least time t given uniform initial conditions in the chaotic part of the cap. The stickiness is sensitively dependent on the radius of the stem r via the Diophantine properties of rho = (2/pi) arccos r. Almost all rho give rise to families of marginally unstable periodic orbits (MUPOs) where P(t) ~ C/t, dominating the stickiness of the boundary. After characterising the set for which rho is MUPO-free, we consider the stickiness in this case, and where rho also has continued fraction expansion with bounded partial quotients. We show that t^2 P(t) is bounded, varying infinitely often between values whose ratio is at least 32/27. When rho has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a log-periodic function. In general, we expect less regular behaviour, with upper and lower exponents lying between 1 and 2. The results may shed light on the parameter dependence of boundary stickiness in annular billiards and generic area preserving maps.

We are interested in the counting process of visits to a small set, and more precisely in its behaviour as the measure of the set goes to 0. We prove the convergence in distribution of this process to a Poisson process under general assumptions. We apply our general results in dierent expanding/hyperbolic contexts. We study in particular the case of chaotic billiard ows: Sinai billiard, the Bunimovich billiard in a stadium, billiards with corners and without cusps. We obtain a general result for visits to balls around a generic point when the system is modeled by a Markov Young tower, and also for visits to balls around an hyperbolic periodic point. This is a joint work with Benot Saussol, inspired by a question by Domokos Szasz and Peter Imre Toth.

Erdos-Renyi laws concern the almost sure behavior of time averages over time windows of varying length in a stationary time series. We discuss recent results on Erdos-Renyi laws for chaotic dynamical systems and their relation to rate of decay of correlations and local large deviations. Some of this work is joint with Nicolai Haydn (University of Southern California), Holger Kantz and Mozhdeh Massah (Max Planck Institute for Complex Systems, Dresden, Germany).

The operator renewal-type approach to obtain polynomial mixing rates in various innite measure-preserving dynamical systems requires that the tails of a certain inducing scheme have regular variation. An almost Anosov dieomorphism is a dieomorphism (in our case on the 2-torus) that satises the Anosov properties except at a nite set of neutral saddle points. In this invertible setting, the regular varia- tion of the tails has been treated only in very specic settings and/or with unsatisfactory estimates. In this talk I want to present a new methods which works in much greater generality and gives much more precise estimates. The mixing results additionally require the use of an anisotropic Banach space of distribution similar to the one used before by Demers & Liverani and by Liverani & Terhesiu. This is joint work with Dalia Terhesiu.

I show that on a compact hyperbolic surface, the mass of an L2- normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassi- cal version, has many applications including control for the Schrdinger equation and the full support property for semiclassical defect mea- sures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.

I will survey old and new results on the Boltzmann-Grad limit of the Lorentz gas. I will in particular focus on recent results with Andreas Strombergsson (Uppsala) for the case of quasicrystalline scatterer con- gurations, and also discuss the Boltzmann-Grad limit for quantum transport in the case of smooth periodic potentials, which is joint work with Jory Grin (Queens University).

A random dynamical system is said to be time-reversible if the sta- tistical properties of orbits do not change after reversing the arrow of time. The degree of irreversibility is captured by the notion of en- tropy production rate. A general formula for entropy production will be presented that applies to a class of thermal perturbations of bil- liard systems, for which it is meaningful to talk about energy exchange between billiard particle and boundary. This formula establishes a re- lation between the purely mathematical concept of entropy production rate and the physical concept of thermodynamic entropy. In particular, it recovers Clausius formulation of the second law of thermodynamics: the system must evolve so as to transfer energy from hot to cold. Fur- ther connections with stochastic thermodynamics will be illustrated with examples of simple "billiard thermal engines." This is joint work with Tim Chumley.

In order to obtain a good statistical theory for a system with a hole in it, the heuristic is that the (exponential) speed of mixing must dominate the (exponential) rate at which mass leaks from the system: so the hole must be appropriately `small'. I'll present joint work with Mark Demers where we analysed this idea for a simple class of systems (Manneville-Pomeau maps with certain `geometric' equilibrium states), giving a complete picture of how the competition between mixing and escape lead to different statistical behaviour. We show a transition from the usual picture of good statistical properties, through a (non-trivial) zone where mixing and escape match exactly, with a terminal transition to subexponential mixing.

We study ergodic properties caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy, topological entropy and pressures, and prove the corresponding variational principles. For unstable metric entropy we obtain affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem. We also obtain existence of Gibbs u-states, differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Frechet differentiability. The results are based on joint works with Weisheng Wu, Yujun Zhu and Yongxia Hua.

We consider a model of 1D multimodal circle maps with strong expansion on most of phase space, including, e.g., the one-parameter family $f_a(x) = L \sin x + a$ for $a \in [0,1)$ with fixed $L >> 1$. Even when L is quite large, the problem of deciding the asymptotic regime (stochastic versus regular) of $f_a$ for a given $a$ involves infinite-precision knowledge of infinite trajectories: outside special cases, this problem is typically impossible to resolve from any checkable finite-time conditions on the dynamics of $f_a$. We contend that the corresponding problem for (possibly quite small) IID random perturbations of the $f_a$ is far more tractable. In our model, we perturb $f_a$ at each timestep by an IID uniformly distributed random variable in the interval $[- \epsilon, \epsilon]$ for a fixed (yet arbitrarily small) $\epsilon > 0$. We obtain a checkable condition, involving finite trajectories of $f_a$ of length ~ $\log(\epsilon^{-1})$, for this random composition to admit (1) a unique, absolutely continuous stationary ergodic measure and (2) a Lyapunov exponent of size approximately $\log L$. Joint with Yun Yang.

We study the properties of €˜infinite-volume mixing€™ for certain classes of expanding one-dimensional maps with indifferent fixed points, preserving an infinite measure. These include the Farey map and the Boole transformation. In particular we focus on the property called €˜global-local mixing€™, which amounts to the decorrelation of a global and a local observable. This property leads to curious limit theorems, which are peculiar to maps with â€strongly neutral€™ fixed points. Joint work with C. Bonanno and P. Giulietti.

The spectrum of transfer operators contains key information on statistical properties of hyperbolic dynamical systems, but only if the operator is acting on an appropriate Banach space. In the past 15 years, dynamicists (and more recently the semiclassical community) have introduced several types of anisotropic spaces of distributions suitable for this purpose. I will start with a brief tour in the jungle of these spaces in order to motivate the introduction of a new norm which combines desirable features of previously existing ones. (In particular, characteristic functions of tame domains are bounded multipliers.)

The classical Lorenz attractor (Lorenz 1963) satisfies various statistical properties such as existence of an SRB measure, central limit theorems, and exponential decay of correlations. The main ingredients are that the attractor is singularly hyperbolic with a $C^r$ stable foliation for some $r>1.$ Certain classes of Lorenz attractors have been obtained analytically for the extended Lorenz equations by Dumortier, Kokubu \& Oka, and more recently by Ovsyannikov \& Turaev. These attractors are singularly hyperbolic but do not have a smooth stable foliation. The aim in this talk (joint work with Vitor Araujo) is to consider statistical properties for singular hyperbolic attractors that do not have a smooth stable foliation. It turns out that existence of an SRB measure, central limit theorems, and mixing hold as in the classical case. But exponential decay of correlations looks currently hopeless. Proving rates of mixing (eg superpolynomial decay) looks perhaps a bit less hopeless

Following on from the pioneering work of Bunimovitch et al on escape rates for the doubling map, it is interesting to consider escape rates for the Gauss map (or continued fraction transformation). In particular, we will discuss the practical issue of computing the escape rate for the interval $[0, 1/n],$ $n >2.$

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.