arXiv subjects

Yuri Lima

Publications and source records attributed to Yuri Lima.

31 records · Page 2Linked to original sources

arXiv2016

Symbolic dynamics for three dimensional flows with positive topological entropy

We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+ε}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface 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.

math.DS↗

arXiv2016

Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. 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 $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.

math.DS↗

arXiv2015

Ergodic properties of equilibrium measures for smooth three dimensional flows

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.

math.DS↗

arXiv2014

Graph-based Polya's urn: completion of the linear case

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. Previous works proved that when $G$ is not balanced bipartite, 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 $\mathcal J(G)$.

math.PR↗

arXiv2013

A generalized Polya's urn with graph based interactions

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 a>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 a<1 then there is a single point v=v(G,a) with nonzero entries such that the proportion converges to v almost surely. A special case is when G is regular and a is at most 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.

math.PR↗

arXiv2013

A Marstrand theorem for subsets of integers

We propose a counting dimension for subsets of Z and prove that, under certain conditions on two such subsets E and F, for Lebesgue almost every real λ the counting dimension of E+[λF] 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+[λF] has positive upper Banach density for Lebesgue almost every λ. The result has direct consequences when E,F are arithmetic sets, e.g. the integer values of a polynomial with integer coefficients.

math.DS↗

arXiv2013

Annihilation and coalescence on binary trees

An infection spreads in a binary tree 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 p=(p_1,...,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 whose only one of the children is infected are infected by this state. In this note we characterize, for every p, the limiting distribution at the root node of the tree as the height 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 p and q, the limiting distribution at the root node of the tree as the height 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.

math.DS↗

arXiv2013

Ergodic properties of skew products in infinite measure

Let (Ω,μ) be a shift of finite type with a Markov probability, and (Y,ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a measure-preserving automorphism of (Y,ν). We study skew products of the form (ω,y) --> (σω,Φ_{ω_0}(y)), where σ=shift map on (Ω,μ). We prove that, when the skew product is conservative, it is ergodic if and only if the Φ_i's have no common non-trivial invariant set. In the second part we study the skew product when Ω={0,1}^Z, μ=Bernoulli measure, and Φ_0,Φ_1 are 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.

math.DS↗

arXiv2012

Law of large numbers for certain cylinder flows

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 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.

math.DS↗

arXiv2012

An Abramov formula for stationary spaces of discrete groups

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 induced 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.

math.DS↗

arXiv2011

Yet another proof of Marstrand's Theorem

In a paper from 1954 Marstrand proved that if K is a Borel subset of the plane with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a new proof of this theorem, extending the techniques developed in a previous work.

math.DS↗

arXiv2010

A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets

In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff dimension greater than 1, then its one-dimensional projection has 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+a}, a>0, for which the sum of their Hausdorff dimension is greater than 1.

math.DS↗

arXiv2010

Z^d-actions with prescribed topological and ergodic properties

We extend constructions of Hahn-Katznelson and Pavlov to Z^d-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build Z^d-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

math.DS↗