In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.

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.

We study a compactification of the space of invariant probability measures for a transitive countable Markov shift. We prove that it is affine homeomorphic to the Poulsen simplex. Furthermore, we establish that, depending on a combinatorial property of the shift space, the compactification contains either a single new ergodic measure or a dense set of them. As an application of our results, we prove that the space of ergodic probability measures of a transitive countable Markov shift is homeomorphic to $\ell_2$, extending to the non-compact setting a known result for subshifts of finite type. Additionally, we explore implications for thermodynamic formalism, including a version of the dual variational principle for transitive countable Markov shifts with uniformly continuous potentials.

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 ‐finite, infinite, ergodic and conservative invariant measure is bounded from above by the entropy at infinity of the geodesic flow.

In this paper, we study aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. In topological dynamics, 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 no 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 behavior 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.

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok's entropy formula in this non-compact setting.

In this note we prove two related results. First, we show that for certain Markov interval maps with infinitely many branches the upper box dimension of the boundary can be read from the pressure of the geometric potential. Secondly, we prove that the box dimension of the set of iterates of a point in H^n with respect to a parabolic subgroup of isometries equals the critical exponent of the Poincare series of the associated group. This establishes a relationship between the entropy at infinity and dimension theory.

It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular, we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite entropy non-locally compact transitive countable Markov shifts and to every locally compact transitive countable Markov shift. In order to prove these results we introduce a topology on the space of measures that generalizes the vague topology to a class of non-locally compact spaces, the topology of convergence on cylinders. We also prove analogous results for suspension flows defined over countable Markov shifts.

We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive entropy diffeomorphisms on compact manifolds, are given. We also discuss the related problem of existence of measures of maximal entropy.

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, 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 determined 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 us, in particular, to compute the entropy of the geodesic flow at infinity.

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 hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups.

We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function $h_\mu(\epsilon,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of $\mu$. Our formulation has the advantage of being, in the authors opinion, more natural when doing computations. As a corollary we obtain a proof of the standard variational principle. We also obtain some relations between the rate distortion function with our function $\tilde{h}_\mu(\epsilon,T,\delta)$, a modification of $h_\mu(\epsilon,T,\delta)$ when replacing the dynamical metrics with the average dynamical metrics. Using our methods we also reprove the main result in [LT]. We will explain how to construct homeomorphisms on closed manifolds with maximal metric mean dimension. We end this paper with some questions that naturally arise from this work.

We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass phenomenon and the measure theoretic entropy on finite volume nonpositively curved manifolds satisfying the Visibility axiom. We provide a general criterion for the same relation to hold between the escape of mass and the measure theoretic entropy. This gives a criterion for the existence of measures of maximal entropy for the geodesic flow on some nonpositively curved manifolds. Finally, we prove some results in the context of countable Markov shifts.

In this paper we study the ergodic theory and thermodynamic formalism of the geodesic flow on non-compact pinched negatively curved manifolds. We consider two notions of entropy at infinity, the topological and the measure theoretic entropy at infinity. We prove that both notions coincide and we relate this quantity with the upper semicontinuity of the entropy map. This relationship was already studied in [RV] for geometrically finite manifolds; in this paper we extend such results to arbitrary pinched negatively curved manifolds. We obtain several applications to the existence and stability of equilibrium states. Our approach has the advantage to include certain uniformly continous potentials, a situation that can not be studied through the methods from [PPS]. Of particular importance are potentials that vanish at infinity and those that satisfy a critical gap condition, that is, strongly positive recurrent potentials. We also prove some equidistribution and counting results for closed geodesics (beyond the geometrically finite case). In particular we obtain a version of the prime geodesic theorem for strongly positive recurrent manifolds.

In this paper we study the phenomenon of phase transitions for the geodesic flow on some geometrically finite negatively curved manifolds. We define a class of potentials going slowly to zero through the cusps of M for which the pressure map exhibits a phase transition. By a careful choice of the metric at the cusp we construct a geometrically finite manifold for which the geometric potential (or unstable Jacobian) exhibits a phase transition. Our results apply, in particular, to the geodesic flow on an M-puncture sphere, for every M≥ 3, and a suitable choice of Riemannian metric.