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Felipe Riquelme

Publications and source records attributed to Felipe Riquelme.

10 recordsLinked to original sources
arXiv2026

Equidistribution and thermodynamics at infinity

We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.

math.DS↗
arXiv2026

On linear escape and the dimension of limit sets in variable negative curvature

In 2004, Bishop proved that for Kleinian groups acting on hyperbolic space, the Hausdorff dimension of the limit set is completely determined by two extremal dynamical behaviors: recurrent geodesics and geodesics escaping linearly to infinity. In this paper, we extend this phenomenon to arbitrary discrete groups of isometries of complete simply connected Riemannian manifolds with pinched negative sectional curvatures $-b^2\leq k\leq -1$. More precisely, we show that the Hausdorff dimension of the limit set coincides with the maximum of the Hausdorff dimensions of the radial limit set and the linear escape limit set.

math.DS↗
arXiv2025

On the Hausdorff dimension of geodesics that diverge on average

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian manifold. Furthermore, we prove that the entropy of a $σ$-finite, infinite, ergodic and conservative invariant measure is bounded from above by the entropy at infinity of the geodesic flow.

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arXiv2023

Pressure at infinity and strong positive recurrence in negative curvature

In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show in particular that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.

math.DG↗
arXiv2022

Exceptional sets for geodesic flows of noncompact manifolds

For a geodesic flow on a negatively curved Riemannian manifold $M$ and some subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set of points whose $ω$-limit do not intersect $A$. We show that if the topological $\ast$-entropy of $A$ is smaller than the topological entropy of the geodesic flow, then the limit $A$-exceptional set has full topological entropy. Some consequences are stated for limit exceptional sets of invariant compact subsets and proper submanifolds.

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arXiv2020

Ergodic optimization and zero temperature limits in negative curvature

In this paper we study aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. It is a well known fact that every continuous potential on a compact metric space has a maximizing measure. Unfortunately, for non-compact spaces this fact is not longer true. For the geodesic flow we provide a criterion that ensures the existence of a maximizing measure for uniformly continuous potentials. We prove that the only obstruction to the existence of a maximizing measure is the full escape of mass phenomenon. To the best of our knowledge, this is the first general result on the existence of maximizing measures for non-compact topological spaces which does not require the potential to be coercive. We study zero temperature limits of equilibrium measures for a suitable family of potentials. We prove some convergence and divergence results for the limiting behaviour of such measures. Among some consequences we obtain that the geodesic flow has the intermediate entropy property and that equilibrium states are dense in the space of invariant probability measures.

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arXiv2018

Ruelle's inequality in negative curvature

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

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arXiv2016

Escape of mass and entropy for geodesic flows

In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing mass, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determinated by the maximal parabolic critical exponent. We also study the pressure of positive Hölder continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows, in particular, to compute the entropy at infinity of the geodesic flow.

math.DS↗
arXiv2016

Entropy in the cusp and phase transitions for geodesic flows

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups.

math.DS↗