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Yuri Lima

Publications and source records attributed to Yuri Lima.

At least 19 recordsLinked to original sources
arXiv2026

Symbolic dynamics for non-uniformly hyperbolic flows

We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $\chi>0$, we code a set which has full measure for every $\chi$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of H\"older continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

math.DS↗
arXiv2025

Superdiffusive central limit theorems for geodesic flows on nonpositively curved surfaces

We prove a nonstandard central limit theorem and weak invariance principle, with superdiffusive normalisation $(t\log t)^{1/2}$, for geodesic flows on a class of nonpositively curved surfaces with flat cylinder. We also prove that correlations decay at rate $t^{-1}$. An important ingredient of the proof, which is of independent interest, is an improved results on the regularity of the stable/unstable foliations induced by the Green bundles.

math.DS↗
arXiv2025

Uniqueness of the measure of maximal entropy for geodesic flows on surfaces

We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the uniqueness of the measure of maximal entropy in this context, as well as it applies to new examples such as the ones constructed by Donnay and Burns-Donnay. We also prove that, in the above context, there is at most one SRB measure.

math.DS↗
arXiv2025

Symbolic dynamics for non-uniformly hyperbolic flows in high dimension

We construct symbolic dynamics for flows with positive speed in any dimension: for each $\chi>0$, we code a set that has full measure for every invariant probability measure which is $\chi$--hyperbolic. In particular, the coded set contains all hyperbolic periodic orbits with Lyapunov exponents outside of $[-\chi,\chi]$. This extends the recent work of Buzzi, Crovisier, and Lima for three dimensional flows with positive speed. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts, and prove that each such class has at most one probability measure that maximizes the entropy.

math.DS↗
arXiv2025

Nonstandard functional central limit theorem for nonuniformly hyperbolic dynamical systems, including Bunimovich stadia

We consider a class of nonuniformly hyperbolic dynamical systems with a first return time satisfying a central limit theorem (CLT) with nonstandard normalisation $(n\log n)^{1/2}$. For such systems (both maps and flows) we show that it automatically follows that the functional central limit theorem or weak invariance principle (WIP) with normalisation $(n\log n)^{1/2}$ holds for H\"older observables. Our approach streamlines certain arguments in the literature. Applications include various examples from billiards, geodesic flows and intermittent dynamical systems. In this way, we unify existing results as well as obtaining new results. In particular, we deduce the WIP with nonstandard normalisation for Bunimovich stadia as an immediate consequence of the corresponding CLT proved by B\'alint & Gou\"ezel.

math.DS↗
arXiv2025

Equilibrium states by synchronization, symbolic extensions, and factors

We combine the two classical topological concepts, time-preserving topological factors and synchronizing time-changes of a continuous flow, and explore some of their thermodynamic consequences. Particular focus is put on equilibrium states and, in particular, measures of maximal entropy, with emphasis on geodesic flows on rank-one surfaces of nonpositive curvature and their time-preserving expansive topological factors for which we investigate the scaled geometric potentials.

math.DS↗
arXiv2024

Measures of maximal entropy for non-uniformly hyperbolic maps

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic 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 (2025), and Viana maps (1997).

math.DS↗
arXiv2024

Homoclinic classes of geodesic flows on rank 1 manifolds

Given a $C^{1+\beta}$ flow $\varphi$ with positive speed on a closed smooth Riemannian manifold, we code two homoclinically related $\varphi$-invariant probabilities by an irreducible countable topological Markov flow. As an application, we give proofs 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.

math.DS↗
arXiv2023

P\'olya urns on hypergraphs

We study P\'olya 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.

math.DS↗
arXiv2023

Lyapunov exponents and nonadapted measures for dispersing billiards

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.

math.DS↗
arXiv2023

Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows

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 $\chi$ 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.

math.DS↗
arXiv2021

Polynomial decay of correlations for nonpositively curved surfaces

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.

math.DS↗
arXiv2020

Symbolic dynamics for nonuniformly hyperbolic maps with singularities in high dimension

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.

math.DS↗
arXiv2020

Symbolic dynamics for nonuniformly hyperbolic systems

This survey describes the recent advances in the construction of Markov partitions for nonuniformly hyperbolic systems. One important feature of this development comes from a finer theory of nonuniformly 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.

math.DS↗
arXiv2020

Isolated photon production and pion-photon correlations in high-energy $pp$ and $pA$ collisions

A phenomenological study of the isolated photon production in high energy $pp$ and $pA$ collisions at RHIC and LHC energies is performed. Using the color dipole approach we investigate the production cross section differential in the transverse momentum of the photon considering three different phenomenological models for the universal dipole cross section. We also present the predictions for the rapidity dependence of the ratio of $pA$ to $pp$ cross sections. As a further test of the formalism, for different energies and photon rapidites we analyse the correlation function in azimuthal angle $Δϕ$ between the photon and a forward pion. The characteristic double-peak structure of the correlation function around $Δϕ\simeq π$ observed previously for Drell-Yan pair production is found for isolated photon emitted into the forward rapidity region which can be tested by future experiments.

hep-ph↗
arXiv2019

Symbolic dynamics for one dimensional maps with nonuniform expansion

Given a piecewise $C^{1+β}$ 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, we code the lift of these measures in the natural extension of the map.

math.DS↗