# «Talks: Monday F. Ledrappier Entropy for compact manifolds: properties, regularity, rigidity Abstract: We consider the stochastic entropy for the ...»

Probability in Dynamics

Abstracts

Rio de Janeiro, May 2630 2014

Talks: Monday

F. Ledrappier Entropy for compact manifolds: properties, regularity, rigidity

Abstract: We consider the stochastic entropy for the Brownian motion on the universal cover of a

compact Riemannian manifold M. We discuss the relation with the other global invariants, the rigidity

if M has no focal points and the regularity under (conformal) changes if M has negative curvature.

P. Nandori Non equilibrium density proles in Lorentz tubes with thermostated boundaries Abstract: We consider a long Lorentz tube with absorbing boundaries. Particles are injected to the tube from the left end. We compute the equilibrium density proles in two cases: the semi-innite tube (in which case the density is constant) and a long nite tube (in which case the density is linear).

In the latter case, we also show that convergence to equilibrium is well described by the heat equation.

In order to prove these results, we obtain new results for the Lorentz particle which are of independent interest. First, we show that a particle conditioned not to hit the boundary for a long time converges to the Brownian meander. Second, we prove several local limit theorems for particles having a prescribed behavior in the past. This is a joint work with Dmitry Dolgopyat.

J. Rousseau Hitting time statistics for observations of dynamical systems Abstract: In this talk, we will study the distribution of hitting and return times for observations of dynamical systems. We will apply this results to get an exponential law for the distribution of hitting and return times for rapidly mixing random dynamical systems. In particular, it allows us to obtain an exponential law for random toral automorphisms, random circle maps expanding in average and randomly perturbed dynamical systems.

M. Clay Rényi's Parking Problem Revisited Abstract: Rényi's parking problem (or 1D sequential interval packing problem) dates back to 1958, when Rényi studied the following random process: Consider an interval I of length x, and sequentially and randomly pack disjoint unit intervals in I until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of I is M (x), so that the ratio M (x)/x is the expected lling density of the random process. Following recent work by Gargano et al., we studied the discretized version of the above process by considering the packing of the 1D discrete lattice interval {1, 2,..., n + 2k − 1} with disjoint blocks of (k + 1) integers but, as opposed to the mentioned result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of r-gaps (0 ≤ r ≤ k) between neighboring blocks.

We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting lling density of the long line segment (as n → ∞) is Rényi's famous parking constant, 0.7475979203....

M. Lenci Random walk in random environment: a dynamicist's approach Abstract: In this talk I will present a case in which ideas and techniques from dynamical systems can say something new about certain popular systems in probability theory. The case is in the area of random walks in random environments (RWREs). A basic result that one wants to prove for a given RWRE is the ergodicity of the environment viewed form the particle (EVP): this is instrumental in proving other dynamical properties, such as, almost sure recurrence, ballisticity, CLT, Invariance Principle, etc. The ergodicity of the EVP is a classical result (e.g., Koslov 1985) under the assumptions of uniform ellipticity and nearest-neighbor jumps. This means that the probability for the walker to move in every coordinate direction is bounded below by a universal constant. The result is also known for certain systems that are elliptic but not uniformly elliptic (i.e., the lower bound may depend on the position of the walker), but was proved on a case-by-case basis. There are reasons to believe that ellipticity is an unnatural and much too strong hypothesis for RWREs. I will give a theorem which implies the ergodicity of the EVP under very general conditions. In particular, the ellipticity assumption is replaced by a more natural, and in some sense optimal, hypothesis on the transitivity of the walks. The proof is based on a dynamical system which is a "stack" of uncountably many piecewise-linear maps of the interval, thus appearing manageable (and visually intuitive) to a dynamicist. Time permitting, I will mention some very recent results obtained jointly with A. Bianchi, G. Cristadoro and M. Ligabo', about vaguely similar processes, namely, continuous-time RWs between Levy-spaced "scatterers" on the real line. The technical core of the proofs is the same type of dynamical system.

** Talks: Tuesday**

M. E. Vares Escape strategies Abstract: Consider rst passage percolation on Zd with passage times given by i.i.d. random variables with common distribution F. Let tπ (u, v) be the time from u to v for a path π and t(u, v) the minimal time among all paths from u to v. We ask whether or not there exist points x, y ∈ Zd and a semi-innite path π = (y0 = y, y1,... ) such that tπ (y, yn+1 ) t(x, yn ) for all n. Necessary and sucient conditions on F are given for this to occur. When the support of F is unbounded, we also obtain results on the number of edges with large passage time used by geodesics.

**R. Bissacot Phase Transitions in Ferromagnetic Ising Models with magneticelds**

Abstract: We study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic eld of type 1/|x|α. We prove that in dimensions d ≥ 2 for all β (the inverse of the temperature) large enough if α 1 there is a phase transition, while if α 1 there is a unique DLR state. Jointly with Marzio Cassandro (GSSI, L'Aquila), Errico Presutti (GSSI, L'Aquila) and Leandro Cioletti (Unb, Brazil). The paper is available in arxiv.

M. Aytaç Laws of rare events for randomly perturbed dynamical systems Abstract: In this talk, we will introduce Extreme Value Laws (EVL) and Hitting/Return Time Statistics (HTS/RTS) in the context of dynamical systems and show the link between these two concepts. Then, we will explain how we exploit this link to prove laws of rare events for certain randomly perturbed dynamical systems. First, we will consider random perturbations of uniformly expanding systems and show that for additive absolutely continuous (w.r.t. Lebesgue) noise, the limiting distribution is standard exponential. Then, we will extend this result to the maps with non-degenerate singularities. Our main ingredients will be decay of correlations against all L1 observables in a suitable Banach space and the rst return time from a set to itself.

S. Gallo Recent results and old open problems concerning g-measures Abstract: g-measures were introduced by Keane (1972) in the Ergodic Theory literature and correspond to what Onicescu and Mihoc (1935) called "chains with complete connections" in the stochastic processes literature. These objects constitute a natural generalization of Markov chains, because the dependence on the past can be unbounded. Several very basic issues may be addressed in this context, and we will focus on 3 of them in this talk: existence, uniqueness and Gibbsianity.

M. Todd Wild attractors: equilibrium states and transience Abstract: The study of equilibrium states and pressure is best understood for positive recurrent systems. In this talk I will discuss unimodal maps of the interval with a wild attractor which means that Lebesgue measure is dissipative despite the system being topologically mixing. Therefore, the natural potential − log |Df | is transient, which is part of an unusual picture, in terms of pressure and equilibrium states, for the family of geometric potentials −t log |Df |. I will explain how the system moves from positive recurrence, through null recurrence to transience as t varies. The methods used involve inducing and combinatorial arguments to nd conformal measures and densities. This is joint work with Henk Bruin.

Talks: Wednesday S. Vaienti On a few statistical properties of sequential (non-autonomous) dynamical systems Abstract: We consider any concatenation of maps in certain classes and we prove a few result of probabilistic nature on the asymptotic behavior of the orbits: loss of memory, extreme values, invariance principles S. Klein Large deviations type estimates for iterates of linear cocycles Abstract: The purpose of this talk is to describe how large deviations type estimates on quantities associated to iterates of a linear cocycle can be used in an inductive scheme in order to derive continuity properties of the Lyapunov exponents as functions of the cocycle. Moreover, I will give an idea about the method used in obtaining such probabilistic estimates in the case of linear cocycles whose base dynamics are given by a torus translation and whose ber actions depend analytically on the base point. The main tools of this method come from harmonic analysis and potential theory. [This is based on joint work with Pedro Duarte.] P. Runo Averaging principle on product space: application on the topology of submanifolds Abstract: Consider an SDE on a product manifold whose trajectories are constant on the second space. We investigate the eective behaviour of a perturbation of order in the second coordinate.

An average principle is shown to hold such that the second coordinate converges to the solution of a deterministic ODE, according to the average of the perturbing vector eld with respect to invariant measures, as goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system (X.M. Li, Nonlinearity 2008). We apply this result to Brownian motion on compact submanifolds of Euclidean spaces, such that the coecients of the corresponding deterministic ODE's in the vertical coordinate are given by the Euler characteristics of the submanifolds.

Talks: Thursday F. Przytycki Lyapunov spectrum for multimodal maps Abstract: For uniformly hyperbolic maps in dimension 1, dimension spectrum for Lyapunov exponent, regular and irregular parts, can be expressed by Legendre transform of `geometric pressure', namely pressure for the potential −t log |f |. I will extend these results to C 3 multimodal maps f on a nite union A of closed intervals in R, the iteration of f on an invariant subset K of A satisfying Darboux property, topologically transitive of positive entropy, having critical points in K. This is a joint work with Katrin Gelfert and Michal Rams, analogous to an earlier paper on the rational maps of the Riemann sphere, with A being Julia set.

and denote by Mmax (F) the set of maximizing measures (that is, the T -invariant probabilities that attain the above maximum). We generalize for this context the notion of Aubry set, rst introduced in ergodic optimization by Contreras-Lopes-Thieullen, 2001. Our main result states that the Aubry set, denoted by Ω(F), contains the support of any maximizing probabilities: ν ∈ Mmax (F) ⇒ supp ν ⊂ Ω(F). Such a characterization allows us, in particular, to show that there always exists a matrix conguration with interesting recurrent properties verifying the maximum value in denition of joint spectral radius. This is a joint work with João Tiago Assunção Gomes (UNICAMP).

M. Rams Random Palis' problem Abstract: Jacob Palis asked a question: given two dynamically dened Cantor sets, is it (generically) true that if the sum of their Hausdor dimensions is greater than 1 then their algebraic sum contains an interval. It was answered positively by Moreira and Yoccoz.

I will present results on the random version of this problem: given k ≥ 2 randomly generated Cantor sets (percolation type, hence stochastically self-similar) for which the expected sum of their Hausdor dimension is greater than one, their algebraic sum almost surely contains an interval. It is a joint work with Karoly Simon.

P. Varandas Multifractal analysis of the irregular set for weak Gibbs measures Abstract: In this talk we address a multifractal description of the set of Birkho irregular points.

We prove estimates for the topological pressure of the set of points whose Birkho time averages are far from the space averages corresponding to the unique equilibrium state when the dynamical system satises some large deviations property. As a consequence we deduce that despite the set of irregular points for maps with specication have full topological pressure the same does not happen for "typical" subsets of the irregular set. Applications include uniformly hyperbolic maps and ows as well as some non-uniformly hyperbolic maps. (this is a joint work with T. Bomm (UFBA) ) E. Pujals Arnold Diusion and Ergodic properties in innite measure Abstract: We will revisit the classical problem of Arnold diusion for hamiltonian systems and we will relate it with problems about robust transitivity and ergodicity for innite measure. To deal with the last part, we will use probabilistic techniques.

I. Melbourne Sharp polynomial estimates for decay of correlations of continuous time dynamical systems Abstract: We extend results of Gouezel and Sarig from discrete to continuous time, establishing sharp lower bounds for decay of correlations. A key ingredient is a continuous time operator renewal equation relating transfer operators for the ow to the transfer operator for the Poincare map. This is joint work with Dalia Terhesiu.

D. Kwietniak On dynamical systems with the specication-like properties

**Abstract:**

Bowen and Sigmund introduced the notion of specication in order to derive ergodic properties of dynamical systems. Examples include mixing SFT's and socshifts, mixing axiom A dieomorphisms,mixing continuous interval (graph) maps, and geodesic ows on manifolds with negative curvature. Many authors have weakened the specication property to apply similar techniques to a wider range of examples. An (incomplete) list contains such names as Climenhaga, Coudène, Dateyama, Gelfert, Hofbauer, Oliveira, Pster, Varandas, and Yamamoto (and their co-authors).