Dynamical Systems Working Seminars

at Rome “Tor Vergata”, Mathematics Department

and related events

- Moyennisation et homogneneisation dans les systemes deterministes et stochastiques.

**Conference at CIRM, Marseille, France, Monday 11 May 2015/ Fryday 15 My 2105** - On the differentiability of thermodynamical quantities

Augusto Armando de Castro Junior–Federal University of Bahia – UFBA.

**Thursday, 4 December 2014, 3:30 pm, room Dal Passo.**

Abstract: In this talk, after a brief survey on linear response formula results, we will focus in some developments that we have carried out at UFBA. Such developments have implications in the regularity of

Lyapunov exponents, Hausdorff Dimension and Decay and Large Deviations rates of non-uniformly expanding systems.

(Joint work with T. Bomfim and P. Varandas.) - Annealed and quenched central limit theorem for random dynamical systems

Romain Aimino – University of Tor Vergata.

**Thursday, 27 November 2014, 3:30 pm, room Dal Passo.**

Abstract: For random dynamical systems, one can distinguish two kinds of limit theorems: annealed results, which refer to the Birkhoff sums seen as functions of both the phase space variable and the choice of the maps composed, and quenched results, which refer to Birkhoff sums for a fixed, but generic, composition of maps. In this talk, I will describe results about the central limit theorem for random dynamical systems consisting of uniformly expanding maps. In particular, I will show that the annealed central limit theorem is valid for such systems, and I will give a necessary and sufficient condition for its quenched version without random centering to hold. This is a joint work with Matthew Nicol and Sandro Vaienti. - Paris–>Rome

**Monday 3 November 2014**- Isothermal and adiabatic thermodynamic transformations from microscopic dynamics

Stefano Olla, Universite’ Paris Dauphine, Paris, Frances.

**Monday 3 November 2014, 16:00 pm, Aula Dal Passo**, Department of Mathematics, University of Roma ‘Tor Vergata’.

Abstract: Isothermal and adiabatic transformations are the basic thermodynamics transformations composing Carnot cycles. I will illustrate how to obtain them from a microscopic dynamics under a diffusive space-time scaling limit. The deterministic irreversible thermodynamic transformations are given by a set of diffusive equations, and the quasi-static reversible transformations are obtained by a further time scaling for limiting slow variation of the applied force. I will also deal with ‘isothermal’ transitions between stationary non equilibrium states. - Landau damping in the Kuramoto model

Gianbattista Giacomin, Universite’ Paris Diderot, Paris, Frances.

**Monday 3 November 2014, 14:30 pm, Aula Dal Passo**, Department of Mathematics, University of Roma ‘Tor Vergata’.

Abstract: The talk will be about the Kuramoto model of globally coupled phase oscillators, a disordered model with deterministic dynamics that has become the prototype model for synchronization phenomena in large families of interacting « units » (cells, individuals, circuit components,…). After an introduction about motivations, known results and open questions, I will present recent results obtained in collaboration with Bastien Fernandez and David Gerard-Varet on the continuum limit of the model with individual frequencies drawn from a distribution with density of class C^n (n>3). The main result is that the Kuramoto order parameter asymptotically vanishes (with polynomial rate n) for every trajectory issued from sufficiently small C^n perturbation, under a suitable condition on the coupling parameter $K$. Such a condition coincides, for basic examples of frequency distributions, with the classical « K<K_c » condition one finds in the literature. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.

- Isothermal and adiabatic thermodynamic transformations from microscopic dynamics
- Entropy of random walks in hyperbolic groups.

Sebastien Gouezel, Universite’ de Rennes 1

**Thursday 30 Ottobre 2014, 3:00 pm, Aula Dal Passo**, Department of Mathematics, University of Roma ‘Tor Vergata’.

Abstract: There are two natural ways to construct random elements in a finitely generated group: one can either take an element uniformly in a big ball, or follow a random walk during a long time. The first process is more related to the geometry of the group, while the second one is much easier to implement from a computational point of view. It is therefore desirable to construct specific random walks for which the two processes give essentially the same result. I will explain why, in a large class of groups (called hyperbolic groups), this is in general impossible. - Slow convergence to non-equilibrium stationary states for coupled rotors.

Christophe Poquet, Universita’ di Roma Tor Vergata

**Thursday 9 Ottobre 2014, 2:30 pm, Aula D’Antoni**, Department of Mathematics, University of Roma ‘Tor Vergata’.

Abstract: We will consider a chain composed of three coupled rotors, attached to thermal baths (with possibly different temperatures) at each extremity. An important feature of this system is that when the middle rotors oscillates rapidly, the energy of this rotor decreases very slowly, due to averaging phenomena. We will construct an effective dynamics for the middle rotor, using averaging techniques, and deduce a family of Lyapunov functions which will allow us to prove the ergodicity of the process with arbitrary polynomial rates.

This is a work in collaboration with N. Cuneo and J.-P. Eckmann. - Systems of isometries and the Rauzy gasket.

Alexandra Skripchenko – Institut de Mathématiques, Université Paris VII

**Thursday, 25 September 2015**, 2:30 pm, room D’Antoni

Abstract: The notion of systems of isometries was introduced by G. Levitt, D. Gaboriau and F. Paulin in 1994 as a natural generalization of interval exchange transformations. In my talk I will consider a special 2-dimensional family of these systems called the Rauzygasket that appeared several times in symbolic dynamics and geometric group theory with different contexts.For this family we will discuss the main dynamical properties like minimality and ergodicity as well as some questions related to invariant measures and Lyapunov exponents. Our main tools include a construction of the suspension flow for systems of isometries, exponential decay and thermodynamical formalism for the related Markov map.

It is a joint work with Artur Avila and Pascal Hubert. - Quasistatic dynamical systems

Mikko Stenlund, Helsinki University

**Thursday, 20 May 2014**, 2:00 pm, room D’Antoni

Abstract: We introduce the notion of a quasistatic dynamical system (QDS), inspired by the namesake processes in thermodynamics. The latter are idealized processes in which the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. We then prove limit theorems for a paradigmatic QDS consisting of strongly chaotic expanding circle maps. (This is joint work with Neil Dobbs.) - Chaos at Tor Vergata (2-3 April 2014)

A two days Conference organized by C.Liverani and S.Luzzatto. -
- Partially hyperbolic systems close to a trivial estension

C. Liverani, Tor Vergata, Rome.**Wednesday, 2 April**10:00 am, Aula De Blasi.

Abstract: I will describe how various limit theorems can be use toward establishing statistical properties of a special class of partially hyperbolic systems. - Unique integrability for C^1 dominated splittings

S. Tureli, ICTP, Trieste.**Wednesday, 2 April**10:30 am, Aula De Blasi.

Abstract: We prove some sufficient conditions for the integrability of C^1 distributions which arise as invariant subbundles of a diffeomorphism. The conditions depend on what are called the singular values of the iterations of the differential and are weaker than the previously known sufficient conditions. - Unique integrability for partially hyperbolic diffeomorphisms on 3 manifolds

K. War, ICTP, Trieste.**Wednesday, 2 April**11:00 am, Aula De Blasi.

Abstract: I will show that partially hyperbolic diffeomorphisms on 3-manifold have integrable central bundle which is to say that there exists a foliation tangent to the central bundle. - Banach spaces for dispersing billiards

Mark Demers, Farfield University, USA.**Wednesday, 2 April**12:00 am, Aula Dal Passo.

Abstract: I will review some basic techniques in the study of transfer operators in dynamical systems, including quasi-compactness and the importance of a spectral gap. I will then describe a good Banach space setting in which the transfer operator associated with a dispersing billiard has recently been proven to enjoy a spectral gap. I will outline some of the consequences of this spectral gap and discuss the application of this technique to some perturbations of dispersing billiards. This is joint work with H.-K. Zhang. - Partially hyperbolic skew products with interval fibers

D. Volk, Tor Vergata, Rome.**Thursday, 3 April**10:30 am, Aula Dal Passo.

Abstract: As a step to understand partially hyperbolic dynamical systems, we studied skew products over Markov shifts with one-dimensional interval fibers. A generic system of this class appeared to be similar to the direct product of the Markov shift and a generic diffeomorphism of the unit interval. In particular, it has finitely many attractors and repellers, and each of them supports an hyperbolic SRB measure. - Decay of correlations for product systems

M. Ruziboev, ICTP, Trieste.**Thursday, 3 April**11:00 am, Aula Dal Passo.

Abstract: We study the rates of decay of correlations for direct products of nonuniformly expanding systems. We show that if components admit a Gibbs-Markov geometrical structure then such a structure can be constructed for the product thus leading to bounds on the decay of correlations in terms of the rates of decay of the components. - SRB measures for partially hyperbolic diffeomorphisms

S.Luzzatto, ICTP, Trieste.**Thursday, 3 April**12:00 am, Aula Dal Passo.

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 + E^cu. Assuming 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. - Decay of correlations for partially hyperbolic diffeomorphisms

X. Li, ICTP, Trieste.**Thursday, 3 April**12:30 am, Aula Dal Passo.

Abstract: We study partially hyperbolic sets K on a Riemannian manifold M, whose tangent space splits as TK M = E cu ⊕ E ss , for which the center-unstable direction E cu is non-uniformly expanding on some local unstable disk. We obtain (stretched) exponential Decay of Correlations for such partially hyperbolic attractors. The strategy is through the construction of an inducing scheme with (stretched) exponential decay of recurrence times can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding behavior in the center-unstable direction. - Statistical stability of Lorenz attractors

M. Soufi, ICTP, Trieste.**Thursday, 3 April**15:00 am, Aula Dal Passo.

Abstract: In this talk we consider the robust family of Lorenz attractors. These attractors are chaotic in the sense that they are transitive and have sensitive dependence on the initial conditions. Moreover, they support SRB measures. We show that the SRB measures depend continuously on the dynamics in the weak star topology. In other words, the Lorenz attractors are statistical stable. - Linear response in dynamical systems: an informal survey

Viviane Baladi, ENS, Paris.**Thursday, 3 April**15:30 am, Aula Dal Passo.

Abstract: We recall recent results on linear response or lack of

it in hyperbolic and nonuniformly hyperbolic dynamical

systems, and mention some open problems.

- Partially hyperbolic systems close to a trivial estension
- Concentration inequalities for random and sequential dynamical systems

Romain Aimino – Université du Sud Toulon Var, France.

**Tuesday, 18 February 2014**, 2:30 pm, room D’Antoni. - Microlocal approach to dynamical zeta functions

Maciej Zworski – University Of California, Berkeley

**Tuesday, 17 December 2013**, 2:00 pm, room D’Antoni

Abstract: Dynamical zeta functions of Selberg, Smale and Ruelle are

analogous to the Riemann zeta function with the product over primes

replaced by products over closed orbits of Anosov flows. In 1967 Smale

conjectured that these zeta functions should be meromorphic but

admitted “that a positive answer would be a little shocking”.

Nevertheless the continuation was proved in 2012 by

Giulietti–Liverani–Pollicott. In my talk I will present a proof of

this result obtained by Dyatlov and myself and inspired by a trace

formula of Guillemin and by recent work of Faure–Sjöstrand. It is

based on a simple idea involving wave front sets and propagation of

singularities: we apply methods of microlocal analysis to the

generator of the flow, in particular, propagation of singularities

results due to Duistermaat-Hörmander, Melrose and Vasy. - Anisotropic spaces for discrete time Sinai billiards

Viviane Baladi – University of Copenhagen

**Thursday, 11 July 2013**, 3:00 pm, room D’Antoni

Abstract: I will explain how to modify the construction in my previous works with Gouëzel on piecewise hyperbolic systems to handle singularities arising from billiards. (Work in progress.) - An ergodic theoretic framework for time-dependent dynamical systems

Cecilia González Tokman – University of New South Wales

**Thursday, 23 May 2013**, 3:00 pm, room D’Antoni

Abstract: Recent developments on multiplicative ergodic theorems provide us with a convenient framework for the study of transport phenomena in time-dependent systems, with driving mechanisms allowed to range from deterministic forcing to stationary noise. I will introduce this framework, paying special attention to the setting of transfer operators. I will then present a recent stability result for random invariant densities of piecewise smooth expanding interval maps. (Joint work with G. Froyland and A. Quas.) - Return- and hitting-time processes for measure preserving maps.

Roland Zweimüller – University of Wien

**Thursday, 9 May 2013**, 2:00 pm, room D’Antoni

Abstract: The asymptotic behaviour of return-times and hitting-times (in particular, of their distributions) of small sets in a probability-preserving dyamical system has been studied in great detail and from various points of view. On the abstract side, the relation between first return- and hitting times is completely understood, and so are the classes of possible limit laws. Somewhat surprisingly, it seems that the corresponding questions for the processes of consecutive (i.e. not just the first) return- and hitting times have remained open for a little while. I will answer both of them. - The problem of exponential mixing for the Lorenz attractor

Oliver Butterley – University of Wien

**Thursday, 21 February 2013**, 2:00 pm, room D’Antoni

Abstract: I will describe the Lorenz flow (a three-dimensional flow supporting a two-dimensional strange attractor) and explain the singular hyperbolic nature of the system. 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. - Perturbing Misiurewicz parameters in the exponential family

Neil Dobbs – University of Helsinki

**Thursday, 31 January 2013**, 2:00 pm, room D’Antoni

Abstract: In one-dimensional real and complex dynamics, a map whose post-singular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a (‘chaotic’) non-hyperbolic map, as per Jakobson’s Theorem for unimodal interval maps. This is despite the hyperbolic parameters forming an open, dense set (at least in the interval setting). We shall present some background results and explain why the contrary holds in the complex exponential family z↦λ·exp(z): Misiurewicz maps are Lebesgue density points for hyperbolic parameters. - Limit theorems for toral translations

Dmitry Dolgopyat – University of Maryland – College Park

**Thursday, 24 January 2013**, 2:00 pm, room D’Antoni

Abstract: We study the discrepancy of the number of visits of a Kronicker sequence on a d dimensional torus to nice sets. We are interested in particular in the question how the answer depends on the geometry of the set. This is a joint work with Bassam Fayad. - Breaking of Ergodicity in Expanding Coupled Map Lattices

Bastien Fernandez – Centre de Physique Théorique CNRS Marseille

**Thursday, 17 January 2013**, 2:00 pm, room D’Antoni

Abstract:To identify and to explain phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations of standard models, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require a stability change of the attractor. Yet, recent proofs of co-existence of several phases in specially designed CML (inspired by PCA) work in the expanding regime where all Lyapunov exponents must be positive.In this talk, I will consider a family of CML with piecewise expanding individual map, global interaction and finite number of sites, in the weak coupling regime where the CML is uniformly expanding. I will show (mostly by numerical means) that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with simultaneous breaking of various symmetries, which can be quantified by measuring the amplitude of magnetization-type characteristics.Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles previous ones as it shows that a loss of ergodicity (associated with symmetry breaking) can occur in basic CML, independently of any decay in the Lyapunov dimension.Time permitting, I will also consider repellers of periodic chains of linearly coupled Lorenz-type maps and present rigorous results obtained by means of symbolic dynamics. In particular I will show that, while all symbolic codes are admissible for sufficiently small coupling intensity (=uncoupled regime), when the interaction strength exceeds a threshold, a large bunch of codes is pruned and an extensive decay of the topological entropy follows suit. Moreover, this quantity appears to be continuous at the threshold and remains extensively bounded below by a positive number in a large part of the expanding regime. - Oscillatory motion for the restricted planar circular three

body problem

Marcel Guardia – University of Maryland, College Park

**Thursday, 6 December 2012**, 4:00 pm, room D’Antoni

Note: The talk will be given remotely using Google+; a public live broadcast will be available at this page.

Abstract: In 1980 J. Llibre and C. Simó proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume that the ratio between the masses of the two primaries was exponentially small with respect to the Jacobi constant. In the present work, we generalize their work proving the existence of oscillatory motions for any value of the mass ratio.We show that, for any mass ratio and large enough Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that creates the so called oscillatory orbits. The main achievement is to rigorously prove the transversality of the invariant manifolds without assuming the mass ratio small, since then this transversality can not be checked by using classical perturbation theory respect to the mass ratio. Finally, we show that in a curve in the two dimensional parameter space formed by the mass ratio and the Jacobi constant, the invariant manifolds of infinity undergo a cubic tangency. This is a joint work with P. Martin and T. M. Seara. - Sub-exponential mixing rates for open systems with interacting particles

Tanya Yarmola – Université de Genève

**Thursday, 22 November 2012**, 2:00 pm, room D’Antoni

Abstract: Rigorous derivations of macroscopic heat conduction laws from the microscopic dynamics of mechanical particle systems coupled to heat reservoirs require good mixing properties of the stationary distributions. For many such systems in nonequilibrium, i.e., with two or more unequal heat reservoirs, the proof of the mere existence of stationary distributions is nontrivial due to the non-compactness of the phase space. It is relatively easy to envision scenarios under which particles slow down (freezing) or speed up (heating), which may push initial distributions towards zero or infinite energy levels and ultimately violate convergence.We consider a class of mechanical systems in which particles interact with an ‘energy tank’ represented by a rotating disk anchored at the center. Particles move freely between collisions. When a particle collides with the disk, an energy exchange occurs, in which the particle exchanges the tangential component of its velocity with the angular velocity of the disk and the normal component of the particle’s velocity changes sign. A system in this class is coupled to heat reservoirs set at possibly different temperatures that absorb particles when they collide with the boundaries of the reservoirs and emit new particles according to the Gibb’s distribution corresponding to the temperatures of the reservoirs.We show that a stationary distribution exists, is unique, and is absolutely continuous with respect to the Lebesgue measure. In addition, all initial distributions converge to the stationary distribution and a large subclass of initial distributions does so at sub-exponential rates. The sub-exponential rates of convergence are primarily due to the influence of slow particles on the system. - Rigidity of Birkhoff billiards

Alfonso Sorrentino, University of Rome 3

**Thursday, 8 November 2012**, 2:00 pm, room D’Antoni

Abstract: Despite being conceptually very simple, mathematical billiards presents a very rigid dynamics, which is completely determined by the geometry of the boundary and therein encoded.

Trying to understand the extent of such rigidity and its implications to the dynamics is a formidable task, which lies behind many intriguing questions and conjectures. In this talk I shall discuss the following problem: given two strictly convex billiards whose maps are conjugate near the boundary (i.e. for small angles), how are their shapes related?

In a joint work with Vadim Kaloshin we prove that if the conjugacy is sufficiently smooth, then the two domains must be similar, i.e. they are the same up to a rescaling and an isometry. Time permitting, I shall discuss how this result could be used to try to tackle a very interesting question posed by Guillemin and Melrose: do the lengths of periodic orbits characterize the shape of the billiard domain? - Fat solenoidal attractors

Peyman Eslami, University of Tor Vergata

**Tuesday, 30 October 2012**, 3:30 pm, room D’Antoni

Abstract:We will discuss the following result by M. Tsujii. Consider the dynamical system T:S^{1}×R→S^{1}×R, T(x,y)=(ax,by+f(x)), where a ≥ 2, a^{-1}<b<1 and f is C^{2}(S^{1}). Then the SBR measure for T is absolutely continuous for almost every f. - Dynamics of some piecewise smooth Fermi-Ulam Models

Jacopo De Simoi, University of Tor Vergata

**Thursday, 25 October 2012**, 2 pm, room D’Antoni

Abstract:We find a normal form which describes the high energy dynamics of a class of piecewise smooth Fermi-Ulam ping pong models; depending on the value of a single real parameter, the dynamics can be either hyperbolic or elliptic.

In the first case we prove that the set of orbits undergoing Fermi acceleration has zero measure but full Hausdorff dimension. We also show that for almost every orbit the energy eventually falls below a fixed threshold. In the second case we prove that, generically, we have stable periodic orbits for arbitrarily high energies, and that the set of Fermi accelerating orbits may have infinite measure.

This is a joint work with D. Dolgopyat. - Singular limits of absolutely continuous invariant measures for families of transitive maps

Peyman Eslami, University of Tor Vergata

**Thursday, 11 October 2012**, 2 pm, room D’Antoni

Abstract: We investigate the dependence on the parameters of absolutely continuous invariant measures for a family of piecewise linear piecewise expanding maps. We construct an example to show that the transitivity of the maps

does not imply the convergence of those measures to the absolutely continuous invariant measure for the limit map. When we deal with piecewise expanding maps, we know that for each of them an acim exists, as was proved by Lasota and Yorke. Moreover, if the map is transitive, then this measure is unique (it follows immediately from the results of Li and Yorke). Consider the case when there is an invariant interval such that the trajectory of almost every point falls into this interval, and the map restricted to this interval is transitive. Then there is also a unique acim, and it is supported by this invariant interval. Keller used this property to construct an example (the so called “W map”) in which such an interval exists for some interval of parameters, and as the parameter converges to a limit value, those intervals become shorter and shrink to a point. Then the weak-* limit of acims is a measure concentrated at one point, while the limit map is transitive and has an acim with the support equal to the whole phase space. He conjectured that this is the only mechanism in which the continuity of the acims can be violated. We show that other mechanisms can exist. We will describe Keller’s example, construct our own example, and compute the invariant density and limit measures. Finally, we look what happens if the slopes on laps (intervals of monotonicity) are constant as in Keller’s example.

The presentation is based on a joint work with Michal Misiurewicz. - A local limit theorem for random walks in balanced environments

Mikko Stenlund, University of Tor Vergata and University of Helsinki

**Thursday, 4 October 2012**, 2 pm, room D’Antoni

Abstract: Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems — yielding a Gaussian density multiplied by a highly oscillatory modulating factor — for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, namely the balanced one, in which such a walk is diffusive. The method of proof is, out of necessity, entirely different from the ballistic case. - Analytic horseshoes and their dimension

Mark Pollicott, Warwick University, UK

**Thursday, 20 September 2012**, 2 pm, room D’Antoni

Abstract: One can associate to a Smale horseshoe its Hausdorff dimension, and study its dependence under perturbation. For smooth perturbations this was studied by Mañé using structural stability. We consider the case of analytic perturbations. Our approach uses dynamical zeta functions and the motivation is an application to the dependence of the density of states for discrete Schroedinger equations. - Homogenization of fast-slow deterministic maps and flows

Ian Melbourne, University of Surrey, UK

**Thursday, 28 June 2012**, 2 pm, room D’Antoni

Abstract: I will describe some results on homogenization of fast-slow deterministic maps and flows. A simplifying aspect is that the fast dynamics is uncoupled (so the system is a skew-product, for the moment anyway), but on the other hand no mixing assumptions are made on the fast flow. The results involve the interpretation of certain stochastic integrals. - Stability index for chaotically driven concave maps

Gerhard Keller, University of Erlangen, Germany

**Thursday, 2 May 2012**, 2 pm, room D’Antoni

Abstract: We study skew product systems driven by a generalized baker’s map and with simple concave fibre maps on R₊ like x ↦ rg(θ) tanh(x) where θ is the stable coordinate of the baker’s map and r is a parameter. The fibrewise attractor is the graph of an upper semicontinuous function θ ↦ Φr (θ) ∈ R₊. There are critical parameters r₋ < rc < r₊ such that Φr ≡ 0 for r < r₋ , Φr is smooth and strictly positive for r > r₊ , {Φr = 0} and {Φr > 0} are dense for r₋ < r ≤ r₊, {Φr = 0} has Lebesgue measure 1 for r ≤ rc and Lebesgue measure 0 for r > rc.

More precisely, we determine the Hausdorff dimension of the sets {Φr = 0} and {Φr >0} in terms of the thermodynamic formalism, and finally we evaluate the stability index of Φr at all regular points θ. This index was introduced by Podvigina and Ashwin [Nonlinearity 24 (2011)] to quantify the local extent of basins of attraction. - Dispersing billiards with moving scatterers

Mikko Stenlund, University of Tor Vergata

**Thursday, 22 March 2012**, 2 pm, room D’Antoni

Abstract: We propose a model of dispersing billiards with slowly moving scatterers, in which the locations of the scatterers are shifted a small distance after each collision. We demonstrate exponential loss of memory for the system by a coupling approach. - Continous time renewal processes

Dalia Terhesiu, University of Tor Vergata

**Thursday, 8 March 2012**, 2 pm, room D’Antoni

Abstract: We will talk about continous time renewal processes (with finite and infinite first moments), focusing on the continous version of the

renewal equation. We will also give a rough summary of results on

limit theorems for continous time Markov pocesses that can be

established via continuous renewal theory techniques. - Standard renewal theory: overview of some results in the literature (part II)

Dalia Terhesiu, University of Tor Vergata

**Thursday, 26 January 2012**, 2 pm, room D’Antoni

Abstract: We will give a rough introduction of the setup for standard renewal sequences (with both finite and infinite mean) and provide a rough summary of the main implications to probability theory. - Standard renewal theory: overview of some results in the literature

Dalia Terhesiu, University of Tor Vergata

**Thursday, 19 January 2012**, 2 pm, room D’Antoni

Abstract: We will give a rough introduction of the setup for standard renewal sequences (with both finite and infinite mean) and provide a rough summary of the main implications to probability theory. - A simple approach to rigorous approximation of invariant measures

Stefano Galatolo, University of Pisa

**Thursday, 12 January 2012**, 2 pm, room D’Antoni

Abstract:There are several numerical approaches to the computation of invariant measures and to the simulation of the statistical behavior of dynamical systems. We will be interested to approximations with an explicit estimate on the error. After reviewing some positive and negative previous results on the problem we will consider a general result on the rigorous approximation of fixed points of operators between Banach spaces. This statement is particularly suited for the approximation of invariant measures in dynamical systems and in particular by the Ulam method. We apply this result to implement an algorithm for the rigorous computation of invariant densities of piecewise expanding maps up to some error in the $ L^{1}$ distance. We show how several related computational and numerical issues can be solved and show some computer experiment. Time permitting we will also discuss how this approach can be applied to give approximation of the physical invariant measure for a class of piecewise hyperbolic system. - (Mathematical Physics Seminar)

Sistemi fuori equilibrio: sfere elastiche e conduzione del calore fra termostati

Giovanni Gallavotti, University of Rome La Sapienza, Physics Department

**Thursday 24 February 2011,**2:00pm, room Dal Passo, Mathematics Department, Roma Tor Vergata.

Abstract: Recenti risultati sulla dinamica di sistemi fuori equilibrio portano a affrontare il problema della conduzione del calore in stati stazionari. La prima difficolta` e’ gia` nello scrivere le equazioni da risolvere, cosa che da luogo a discussioni senza conclusioni accettate unanimamente: esporro` un punto di vista e qualche risultato che ne consegue. (collaborazione con G.Gentile e A.Giuliani) - A Dynamics Day in Tor Vergata

**Friday 11 February 2011,**room Dal Passo, Mathematics Department, Roma Tor Vergata.- Exponential decay of correlations for piecewise hyperbolic contact flows

Viviane Baladi Ecole normale supérieure, Paris

**10am-11am**

Abstract: Strong ergodic properties (such as exponential mixing) have been proved for various smooth dynamical systems by first obtaining a spectral gap for a suitable “transfer” operator acting on an appropriate Banach space. Some natural dynamical systems, such as discrete or continuous-time billiards, are only piecewise smooth, and the discontinuities pose serious technical problems in the definition of the Banach norm. In the discrete-time situation, with Sebastien Gouezel (J Mod Dyn 2010), we overcame these problems by using classical tools such as complex interpolation on anisotropic Sobolev-Triebel spaces, and an old result of Strichartz on Fourier multipliers. With Carlangelo Liverani, we now obtain exponential decay of correlation for piecewise hyperbolic contact flows. This is possible by combining the ideas in the work with Gouezel on discrete-time dynamics with techniques developed by Liverani a few years ago (based on ideas introduced by Dima Dolgopyat) to study contact Anosov flows. - Invariant symplectic foliation and reductions of the planetary N-body problem

Luigi Chierchia Universita di Roma 3, Rome

**11:30 am-12:30 pm**

Abstract: By means of new global Darboux coordinates we show that the phase space of the planetary N-body problem is foliated by invariant symplectic leaves corresponding to different orientation of the total angular momentum.On each leaf is also possible to perform esplicitely a total symplectic reduction of rotations. This approach allow to remove well known degeneracies and prove, for instance, Kolmogorov non degeneracy, or to construct Birkhoff normal forms up to any order. Existence of Lagrangian lower-dimensional invariant tori and periodic solutions easily follow. (Joint work with Gabriella Pinzari). - Logarithm laws, decay of correlations and arithmetical properties

Stefano Galatolo, Universita di Pisa, Pisa

**2:00 pm-3:00 pm**

Abstract: It is known that if a dynamics has fast enough decay of correlations then some logarithm law will hold. Logarithm laws consider the behavior of the time which is needed for a typical point to enter in a sequence of decreasing targets. A logarithm law is established when this time increases (having the same scaling behavior) as the inverse of the measure of the targets. We will apply this to Lorenz like flows and the geodesic flows in variable negative curvature obtaining a logarithm law for these systems. - Regularity of the entropy of some random walks

François Ledrappier, University of Notre Dame, IN, USA and CNRS Paris

**3:30 pm-4:30 pm**

Abstract: We consider random walks on a free group. We vary the directing probability among the ones with a fixed generating finite support. We prove that the entropy of the random walk and the linear drift are real analytic functions of the probability.

- Exponential decay of correlations for piecewise hyperbolic contact flows
- On the standard map

Jacopo de Simoi, University of Rome Tor Vergata

**Thursday 27 January 2011,**2pm, room Dal Passo, Mathematics Department, Roma Tor Vergata. - Dolgopyat estimate for flows (an introduction)

Liverani Carlangelo, University of Rome Tor Vergata

**Thursday 20 January 2011,**2pm, room 1101 Mathematics Department,

Roma Tor Vergata.

Abstract: Dolgopyat estimate is the key ingredient to prove exponential decay of correlations for a vast class of hyperbolic flows. I will try to describe how to obtain it and how to use it. - NEW RESULTS IN QUANTUM ERGODIC THEORY

Francesco Fidaleo, University of Rome Tor Vergata

**Monday 20 Dicember 2010,**3pm, room 1101 Mathematics Department,

Roma Tor Vergata. - Dynamical Zeta Functions for Anosov Flows

Paolo Giulietti, University of Rome La Sapienza and University of Rome Tor Vergata

**Wednesday 3 Novembre 2010,**4 pm, room F Mathematics Department,**University of Roma La Sapienza**

Abstract: Given a suitable continuous dynamical system, one can define a zeta function using orbits, similarly to what has been done traditionally with the Riemann zeta function and the prime numbers. In this talk I will present some results, obtained in collaboration with C. Liverani and M. Pollicott, for the Ruelle Zeta function for geodesic flows on manifolds of negative curvature. We are able to prove in this case, and in general for smooth Anosov flows, that the Ruelle zeta function is meromorphic on the whole complex plane. I will present the main elements of the proof i.e. transfer operators on anisotropic spaces and how they are used in computing “ad hoc” traces. - Statistical properties of hyperbolic dynamical systems: a functional approach

Carlangelo Liverani, University of Rome Tor Vergata

**Thursday, 28 October 2010**, 2 pm, room Dal Passo

Abstract: First I will present an overview of the results that can be obtained by the direct study of the Transfer operator on appropriate Banach spaces. In particular: decay of correlations, stability, linear response, limit laws, zeta functions etc.. I will mention the advantages and the shortcoming with respect to different approaches. Then I will discuss some more technical points. - Escaping orbits in a family of anti-integrable limits of the standard map

Jacopo De Simoi, University of Rome Tor Vergata

**Thursday, 21 October 2010**, 2 pm, room Dal Passo

Abstract: The dynamics of a mechanical system undergoing Fermi acceleration can be described by a family of maps which can be viewed as anti-integrable limits of the standard map. In the first part of the talk we will present some results about abundance of so-called escaping orbits, along with results about presence of stable periodic motions (elliptic islands). In the second part we will focus on the techniques used in the proof of the results and present some problems that will be the subject of some future work. - The dynamical systems approach to the study of the joint spectral radius

Morris Ian, University of Rome Tor Vergata

**Thursday, 14 October 2010**, 2 pm, room Dal Passo

Abstract: The joint spectral radius of a finite set of matrices is defined to be the maximum possible exponential growth rate of long products of matrices drawn from that set. Joint spectral radii arise naturally in various mathematical areas including control theory, coding theory, fractal dimension and combinatorics. In this talk we discuss some questions in the theory of the joint spectral radius which can be approached by studying the dynamics of the shift transformation on the space of sequences of matrices. - Macroscopic Laws and Dynamical Systems

Liverani Carlangelo, University of Rome Tor Vergata

**Thursday, 7 October 2010**, 3 pm, room 1101