Speakers

Title: Dirac disintegration and vanishing center exponent

Abstract: An example due to A. Katok and also J. Yorke which appears in a note written by J. Milnor (Fubini foiled: Katok paradoxical example in measure theory) reveals a "pathological" behavior of a foliation by analytics leaves in the unit square: There exists a Lebesgue full measure subset intersecting each leaf at most one point. In other words, Lebesgue measure disintegrates into Dirac masses.Shub and Wilkinson found the same phenomenon for the center foliation of some partially hyperbolic dynamics with one-dimensional center and non-zero center exponent. Pesin and Hirayama studied a higher dimensional version with compact center bundle and positive sum of center exponents.

In all of the above conclusions, the center leaves are compact and there is no room for expansion. In joint work with J. Zhang we prove that for derived from Anosov diffemorphisms (non-compact center leaves) any ergodic measure with zero center Lyapunov exponent disintegrates into Dirac measures.

Title: Statistical properties of (not so) simple partially hyperbolic systems

Abstract: Fast-slow systems are ubiquitous in nature. When the fast dynamics enjoys uniform hyperbolic properties fast-slow systems gives rise, naturally, to interesting examples of partially hyperbolic systems. I will discuss the simplest of such example as it provides both insides to the general problem and open mathematical challenges.

Tittle: Intermingled phenomena for Kan diffeomorphisms

Abstract: In 1994, Ittai Kan provided the first examples of maps with intermingled basins. The maps correspond to skew products on the cylinder, with a uniformly expanding map in the base and having two physical measures supported on the boundary of the cylinder. Two physical measures are intermingled if, for every open set U in the cylinder, each basin has a positive Lebesgue measure on U. Note that the Lebesgue measure on the cylinder plays a role of reference measure in the definition of physical measure and intermingles phenomenon.

Throughout this talk, we will discuss the intermingles phenomenon when the reference measures are other than Lebesgue.

This is a joint work with Bárbara Nuñez Madariaga from PUCV.

Tittle: Robust transitivity and domination for endomorphisms displaying critical points

Abstract: We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows us to get some topological obstructions for the existence of robustly

transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation. This is a joint work with R. Potrie, E. Pujals and W. Ranter.

Tittle: Recent results on the classical and dynamical Markov and Lagrange spectra

Abstract: The classical Lagrange and Markov spectra are closed subsets of the real line consisting of the best constants of Diophantine approximations of certain irrational numbers and indefinite binary quadratic forms, respectively. The Lagrange and Markov dynamical spectra, was introduced by Moreira and share several geometric and topological aspects with the classical ones.

In this talk, we will present recent results on the topological structure of both spectra in the classical and dynamical forms. In particular, we will present Phase Transition theorems for the Markov and Lagrange dynamical spectra which allows us to conjecture a strong statement for the classical spectra. These results are consequences of several collaborations with C. Matheus, C. G. Moreira and S. Vieira.

Title: Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms on T^3 (Part 1)

Abstract: In this talk, we will consider an Anosov diffeomorphism on T^3 admitting a dominated splitting of the form TT^3 = E^s + E^c + E^u, where E^c expands uniformly. There are two expanding foliations that we can consider for this system. The unstable foliation W^cu, which is tangent to E^c+E^u, and the strong unstable foliation W^u, which is tangent to E^u. The unstable foliation is well understood, it is minimal and there is a unique invariant measure whose disintegrations along its leaves are absolutely continuous, the SRB measure. However, the strong unstable foliation is not well understood. In this two part talk, we will study invariant measures whose disintegrations along strong unstable leaves are absolutely continuous. These are the so-called u-Gibbs measures. We will study how they relate to the SRB measure. We find conditions that guarantee that a u-Gibbs measure is SRB. This is a joint work with Sébastien Alvarez and Bruno Santiago.

Title: Intermittency and non-statistical behavior

Abstract: I will present recent and ongoing work, joint with Stefano Luzzatto, in which we study a class F of interval maps with two intermittent fixed points. We show that a dense subclass of F exhibits non-statistical behavior, in particular, we show that every map in this subclass does not have a physical measure.

Title: Rotational Chaos for annular dynamics

Abstract: We study homeomorphisms of surfaces,with emphasys the annulus, trying to characterize when the associated dynamical system has positive topological entropy and rotational entropy. Using topological techniques, we show (with P. Le Calvez) that the presence of any type of shear in conservative settings imply that either the dynamics is near-integrable or it must have a rotational horseshoe. For the general case, we show (with A. Passeggi) that if one can find an invariant circloid with sheared dynamics (having non-trivial rotation set), then the dynamics must also have a rotational horseshoe, solving a conjecture dating back to the 80s.

Title: Hausdorff dimension for invariant measures of circle homeomorphisms with breaks

Abstract: By a classical theorem of Denjoy, any sufficiently regular piece-wise smooth circle homeomorphism with finitely many branches (often called a circle homeomorphism with breaks) and irrational rotation number is topologically conjugated to an irrational circle rotation. In particular, it admits a unique invariant probability measure.

We will discuss dimensional properties of this measure and show that, generically among the circle homeomorphisms with breaks having zero mean nonlinearity (e.g. piece-wise linear), this unique invariant probability measure has zero Hausdorff dimension. This was known to be the case in the non-zero mean nonlinearity setting after a recent work of K. Khanin and S. Kocić.

To encode this generic condition, we consider piece-wise smooth homeomorphisms as generalized interval exchange transformations of the interval and rely on the notion of combinatorial rotation number, which plays an analogous role to rotation number for circle homeomorphisms.

Title: Equidistribution of orbits of rational numbers

Abstract: Every rational number has a finite continued fraction expansion. In 2018, D. Ofir and U. Shapira proved that most orbits of rational numbers, with respect to the Gauss map, equidistribute with respect to the Gauss measure. We will provide an alternative proof of this result that rests on Large Deviations estimates for suspension flows over countable Markov shifts. If time permits, I will state a higher dimensional version of the main result that holds for the Jacobi-Perron map. This is joint work with F. Riquelme and A. Velozo.

Title: On Ergodic Theory of Impulsive Semiflows

Abstract: Impulsive Dynamical Systems (IDS) can be seen as suitable mathematical models of real world phenomena that display abrupt changes in their behaviour. More precisely, an IDS is described by three objects: a continuous semiflow on a space X; a set D contained in X where the flow undergoes sudden perturbations; and an impulsive function from D to X, which determines the change in the trajectory each time it collides with the impulsive set D.

In spite of their great range of applications, IDS have started being studied from the viewpoint of ergodic theory only quite recently in the work of Alves and Carvalho [2014]. A key challenge, inherent to the dynamics, is that in general, an impulsive semiflow is not continuous.

In this talk I will provide sufficient conditions for the existence of invariant measures that imply the ones given by Alves and Carvalho and are somewhat easier to verify. Moreover, I will discuss how typical is the invariance of the non-wandering set of an impulsive semiflow. I will finish the talk with some open problems. This talk is based upon two works with Afonso and Bonotto and with Torres and Varandas.

Title: Continuity of Lyapunov exponents for locally constant cocycles

Abstract: The continuity of Lyapunov exponents has been extensively studied in the context of linear cocycles. However, there are few theorems that provide information for the case of diffeomorphisms. In this talk, we will review some of the known results and explain the main difficulties that appear when trying to adapt the usual techniques to the study of center Lyapunov exponents of partially hyperbolic diffeomorphisms.In this lecture we will present results on the continuity of Lyapunov exponents in the Hölder topology for locally constant cocycles with values in SL(2,R) defined over a Bernoulli shift. In particular, we will prove that the Bocker-Viana discontinuity example is not typical among cocycles with sufficiently small upper Lyapunov exponents. This is a joint work with Catalina Freijo.

Title: Accessibility for dynamically coherent partially hyperbolic diffeomorphism with 2D center

Abstract: The Pugh-Shub accessibility conjecture says that for any integer r greater or equal to 2 stable accessibility is open and dense among the set of C^r partially hyperbolic diffeomorphisms, volume preserving or not.

In a joint work with Martin Leguil, we show that the conjecture is true among the set of stably dynamically coherent partially hyperbolic diffeomorphisms with 2 dimensional center under a strong bunching condition.

Title: Uniqueness of equilibrium states for Lorenz attractors in any dimension

Abstract: We consider the thermodynamic formalism for Lorenz attractors of flows in any dimension. Under a mild condition on the Holder continuous potential function φ, we prove that for an open and dense subset of C1 vector fields, every Lorenz attractor supports a unique equilibrium state. In particular, we obtain the uniqueness for the measure of maximal entropy.

This corresponds to a joint paper with Jiagang Yang and Fan Yang.

Title: Rigidity of u-Gibbs measures near conservative Anosov diffeomorphisms on T^3 (Part 2)

Abstract: In this talk, we will consider an Anosov diffeomorphism on T^3 admitting a dominated splitting of the form TT^3 = E^s + E^c + E^u, where E^c expands uniformly. There are two expanding foliations that we can consider for this system. The unstable foliation W^cu, which is tangent to E^c+E^u, and the strong unstable foliation W^u, which is tangent to E^u. The unstable foliation is well understood, it is minimal and there is a unique invariant measure whose disintegrations along its leaves are absolutely continuous, the SRB measure. However, the strong unstable foliation is not well understood. In this two part talk, we will study invariant measures whose disintegrations along strong unstable leaves are absolutely continuous. These are the so-called u-Gibbs measures. We will study how they relate to the SRB measure. We find conditions that guarantee that a u-Gibbs measure is SRB. This is a joint work with Sébastien Alvarez and Bruno Santiago.

Tittle: Multidimensional rates for a deterministic weak invariance principle

Abstract: A chaotic dynamical system often gives rise to interesting statistical properties, such as the strong law of large numbers and the central limit theorem. In this talk, we will glimpse into the world of smooth ergodic theory, recalling limit theorems for random variables and processes which are generated by a deterministic system. In particular, we will present new rates of convergence to a multidimensional Brownian motion for nonuniformly expanding maps and semiflows. These results were obtained using martingale techniques and a new martingale-coboundary decomposition in the style of Gordin.

Title: A geometric approach to nonexpansive directions

Abstract: In this talk, I will introduce a new framework for studying the notion of nonexpansivity, which is valid for all countable group actions. In particular, I will explain a new result that simultaneously generalizes Schwartzman’s and Boyle-Lind's theorems on nonexpansivity of subspaces. This is based on a joint work with Samuel Petite and Alejandro Maass.