We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.Theorem A. Let M be a closed rank 1 Riemannian manifold without focal points, F be the geodesic flow over M , and ϕ : T 1 M → R be a Hölder potential. If ϕ satisfies the pressure gap condition, then ϕ has a unique equilibrium state µ ϕ .The origin of Theorem A traces back to the work of Bowen [Bow74] where he showed that the bounded distortion property (also known as the Bowen property) on the potential and the expansivity and the specification property on the dynamical system guarantee the existence of a unique equilibrium state. Natural examples of such systems are uniformly hyperbolic systems. Since the work of Bowen, his result has been extended in various directions, and one such direction aims at relaxing the assumptions on the base dynamical systems. Recently, Climenhaga and Thompson [CT16] developed a set of criteria that guarantees the existence of a unique equilibrium state which applies to many non-uniformly hyperbolic systems; see [CFT18], [BCFT18], [CKP18], and [CFT19].Among non-uniformly hyperbolic dynamical systems arising from geometry, the first result in the same flavor as Theorem A was the work of Knieper [Kni98]. He employed Patterson-Sullivan theory to establish the uniqueness of the measure of maximal entropy for geodesic flows over nonpositively curved manifolds. Twenty years later, Burns, Climenhaga, Fisher, and Thompson [BCFT18] extended Knieper's result to equilibrium states for potentials with the pressure gap. Their approach is inspired by the previously mentioned work of Bowen [Bow74]. Previous work by the authors [CKP18] used the same approach to further generalize this result to surfaces without focal points. Finally, in this work, Theorem A shows that the same philosophy holds for higher dimensional manifolds without focal points.From Theorem A, it automatically follows that these equilibrium states are ergodic. It is natural to ask whether such equilibrium states possess stronger ergodic properties. Indeed, for uniformly hyperbolic systems, the unique equilibrium states for Hölder potentials have many stronger ergodic properties; these include being Bernoulli, and having equidistribution property by weighted periodic orbits as well as statistical properties such as the central limit theorem and the large deviation property; see [PP90] and [KH97].The following theorem partially answers the question above, and establishes several ergodic properties of the unique equilibrium state µ ϕ from Theorem A.Theorem B. In the setting of Theorem A, the unique equilibrium state µ ϕ has the following properties: µ ϕ has the K-property and is fully supported. Moreover, µ ϕ is equal to the weak- * limit of the weig...

In this paper, we study an interesting curve, so-called the Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especially representations corresponding to Riemann surfaces with cusps. Using Thermodynamic Formalism (for countable Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Marc Burger [Bur93] and Richard Sharp [Sha98] for convex-cocompact Fuchsian representations.

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichm\"uller theory. This paper is motivated by Pollicott and Sharp's earlier work.Comment: 25 pages and 12 figure

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