With Davi Obata and Mauricio Poletti

Abstract: For C^{1+} maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study the finiteness/uniqueness of such measures in several different settings: finite horizon dispersing billiards, codimension one partially hyperbolic endomorphisms with “large” entropy, robustly non-uniformly hyperbolic volume-preserving endomorphisms as in Andersson-Carrasco-Saghin, and strongly transitive non-uniformly expanding maps.

With Mauricio Poletti

Abstract: We prove that the homoclinic class of every hyperbolic periodic orbit of a geodesic flow over a C^\infty closed rank 1 Riemannian manifold equals the unit tangent bundle. As an application, we give a proof using symbolic dynamics of the theorem of Knieper on the uniqueness of the measure of maximal entropy and theorems of Burns et al on the uniqueness of equilibrium states.

With Pedro Alves and Matheus Barros

Abstract: We study Pólya urns on hypergraphs and prove that, when the incidence matrix of the hypergraph is injective, there exists a point v=v(H) such that the random process converges to v almost surely. We also provide a partial result when the incidence matrix is not injective.

With Jérôme Buzzi and Sylvain Crovisier

Abstract: We construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each \chi>0 , we code a set of full measure for every invariant probability measure which is \chi–hyperbolic. These include all ergodic measures with entropy bigger than χ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of [-\chi,\chi]. This contrasts with a previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts.

With C. Matheus and I. Melbourne.

To appear in Transactions of the AMS.

Abstract: We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows.

With E. Araujo and M. Poletti.

To appear in Memoirs of the AMS.

Abstract: We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.

With Vaughn Climenhaga, Mark Demers and Hongkun Zhang

Commun. Math. Phys. 405, 24 (2024).

Abstract: For hyperbolic systems with singularities, such as dispersing billiards, Pesin theory as developed by Katok and Strelcyn applies to measures that are “adapted” in the sense that they do not give too much weight to neighborhoods of the singularity set. The zero-entropy measures supported on grazing periodic orbits are nonadapted, but it has been an open question whether there are nonadapted measures with positive entropy. We construct such measures for any dispersing billiard with a periodic orbit having a single grazing collision; we then use our construction to show that the thermodynamic formalism for such billiards has a phase transition even when one restricts attention to adapted or to positive entropy measures.

With C. G. Moreira.

Bull. Braz. Math. Soc. (N.S.) 54 (2023), no. 2, Paper No. 28, 4 pp.

Abstract: Given positive integers \ell<n and a real d\in (\ell,n) , we construct sets K\subset R^n with positive and finite Hausdorff d−measure such that the Radon-Nikodym derivative associated to all projections on \ell −dimensional planes is not an L^p function, for all p>1.

Ergodic Theory Dynam. Systems **41** (2021), no. 9, 2591-2658.

Abstract: This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

With L. Backes, M. Poletti, and P. Varandas.

Ergodic Theory Dynam. Systems **40** (2020), no. 11, 2947-2969.

Abstract: We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a u-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched linear cocycles.

Ann. Inst. H. Poincaré C Anal. Non Linéaire **37** (2020), no. 3, 727-755.

Abstract: Given a piecewise C^{1+\beta} map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, for each \chi>0 we construct a finite-to-one Hölder continuous map from a countable topological Markov shift to the natural extension of the interval map, that codes the lifts of all invariant probability measures as above with Lyapunov exponent greater than \chi almost everywhere.

With O. Sarig.

J. Eur. Math. Soc. (JEMS) **21** (2019), no. 1, 199-256.

Abstract: We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C^{1+\epsilon} flows on closed smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C^\inftysurface has at least const\times e^{hT}/T simple closed orbits of period less than T, whenever the topological entropy h is positive – and without further assumptions on the curvature.

With C. Matheus.

Ann. Sci. Éc. Norm. Supér. (4) **51** (2018), no. 1, 1-38.

Abstract: This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities \mathcal D. We allow the derivative of points nearby \mathcal D to be unbounded, of the order of a negative power of the distance to \mathcal D. Under natural geometrical assumptions on the underlying space M, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to \mathcal D. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich billiards.

Israel J. Math. **214** (2016), no. 1, 43-66.

Abstract: Let (\Omega,\mu) be a shift of finite type with a Markov probability, and (Y,\nu) a non-atomic standard measure space. For each symbol i of the symbolic space, let \Phi_i be a non-singular automorphism of (Y,\nu). We study skew products of the form (\omega,y)\mapsto (\sigma\omega,\Phi_{\omega_0}(y)), where \sigma is the shift map on (\Omega,\mu). We prove that, when the skew product is recurrent, it is ergodic if and only if the \Phi_i‘s have no common non-trivial invariant set.

In the second part we study the skew product when \Omega=\{0,1\}^{\mathbb Z}, \mu is a Bernoulli measure, and \Phi_0,\Phi_1 are \mathbb R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.

With F. Ledrappier and O. Sarig.

Comment. Math. Helv. **91** (2016), no. 1, 65-106.

Abstract: Let \{T^t\} be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either \{T^t\} is Bernoulli, or \{T^t\} is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.

Stoch. Dyn. **16** (2016), no. 2, 1660007, 13 pp.

Abstract: Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. This model was introduced by Benaim, Benjamini, Chen, and Lima. When G is not balanced bipartite, Chen and Lucas proved that the proportion of balls in the bins converges to a point w(G) almost surely.

We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point w(G) or a closed interval \mathcal J(G).

With M. Benaïm, I. Benjamini, and J. Chen.

Random Structures Algorithms **46** (2015), no. 4, 614-634.

Abstract: Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power \alpha>0. We characterize the limiting behavior of the proportion of balls in the bins.

The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if \alpha<1 then there is a single point v=v(G,\alpha) with non-zero entries such that the proportion converges to v almost surely.

A special case is when G is regular and \alpha\leq 1. We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.

With I. Benjamini.

Stoch. Dyn. **14** (2014), no. 3, 1350023, 11 pp.

Abstract: An infection spreads in a binary tree \mathcal T_n of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector {\bf p}=({\bf p}_1,\ldots,{\bf p}_{k+1}). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes such that only one of the children is infected are infected by this state. In this paper we characterize, for every {\bf p}, the limiting distribution at the root node of \mathcal T_n as n goes to infinity.

We also consider a variant of the model when k=2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of {\bf p} and q, the limiting distribution at the root node of \mathcal T_n as n goes to infinity.

The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.

With Y. Hartman and O. Tamuz.

Ergodic Theory Dynam. Systems **34** (2014), no. 3, 837–853

Abstract: Let (G,\mu) be a discrete group equipped with a generating probability measure, and let \Gamma be a finite index subgroup of G. A \mu-random walk on G, starting from the identity, returns to \Gamma with probability one. Let \theta be the hitting measure, or the distribution of the position in which the random walk first hits \Gamma. We prove that the Furstenberg entropy of a (G,\mu)-stationary space, with respect to the action of (\Gamma,\theta), is equal to the Furstenberg entropy with respect to the action of (G,\mu), times the index of \Gamma in G. The index is shown to be equal to the expected return time to \Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,\mu), we prove that the random walk entropy of (\Gamma,\theta) is equal to the random walk entropy of (G,\mu), times the index of \Gamma in G.

Ergodic Theory Dynam. Systems **34** (2014), no. 3, 801–825

Abstract: We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit a law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a Gaussian distribution.

With C. G. Moreira.

Combin. Probab. Comput. 23 (2014), no. 1, 116-134.

Abstract: We propose a counting dimension for subsets of \mathbb Z and prove that, under certain conditions on E,F\subset\mathbb Z, for Lebesgue almost every \lambda\in\mathbb R the counting dimension of E+\lfloor \lambda F\rfloor is at least the minimum between 1 and the sum of the counting dimensions of E and F. Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E+\lfloor \lambda F\rfloor has positive upper Banach density for Lebesgue almost every \lambda\in\mathbb R. The result has direct consequences when E,F are arithmetic sets, e.g., the integer values of a polynomial with integer coefficients.

Ergodic Theory Dynam. Systems **32** (2012), no. 1, 191-209.

Abstract: We extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to \mathbb Z^d-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build \mathbb Z^d-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

With C. G. Moreira.

Bull. Braz. Math. Soc. **42** (2011), no. 2, 331-345.

Abstract: In a paper from 1954 Marstrand proved that if K\subset\mathbb R^2 is a Borel set with Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem, extending the techniques developed in our previous paper.

With C. G. Moreira.

Expo. Math. 29 (2011), no. 2, 231-239.

Abstract: In a paper from 1954 Marstrand proved that if K\subset\mathbb R^2 has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^{1+\alpha},\alpha>0, for which the sum of their Hausdorff dimension is greater than 1.

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