Entropy in the cusp and phase transitions for geodesic flows

Abstract

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.

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

“…In turn, Schapira and Tapie were motivated, in part, by work on strongly positive recurrent potentials for countable Markov shifts due to Gurevich-Savchenko [23], Sarig [56,57], Ruette [51], and Boyle-Buzzi-Gómez [8]. Other relevant precursors to our results include the work Iommi-Riquelme-Velozo [24], Riquelme-Velozo [49], and Velozo [65].…”

“…In this paper, we use the Renewal Theorem of Kesseböhmer and Kombrink [30] to establish counting and equidistribution results for well-behaved potentials on topologically mixing countable Markov shifts with (BIP) in the spirit of Lalley's work [33] on finite Markov shifts. Inspired by work of Schapira-Tapie [60,61], Dal'bo-Otal-Peigné [17], Iommi-Riquelme-Velozo [24] and Velozo [65] in the setting of geodesic flows on negatively curved Riemannian manifolds, we define notions of entropy gap at infinity for our potentials. Our results require that the potentials are non-arithmetic, eventually positive and have an entropy gap at infinity.…”

Excursions to the cusps for geometrically finite hyperbolic orbifolds, and equidistribution of closed geodesics in regular covers

“…Einsiedler, Kadyrov and Pohl generalized these results to diagonal actions on spaces Γ\G where G is a connected semisimple real Lie group of rank 1 with finite center, and Γ is a lattice [11]. Finally, Iommi, Riquelme and Velozo (in two papers with different sets of coauthors) considered entropy in the cusp for geometrically finite Riemannian manifolds with pinched negative sectional curvature and uniformly bounded derivatives of the sectional curvature [18,28]. This latter setting is substantially more general than ours, though in the constant curvature case Theorem 1.2 gives more information.…”

Pressure, Poincaré series and box dimension of the boundary

“…If we denote by P T k p¨q the pressure associated to the dynamical system T k then a classical result relates it to the pressure of T (see [Wa,Theorem 9.8 (i)]). Indeed, if f : Λ Ñ R is a locally Hölder potential (see [IRV,p.616] for precise definition) then…”

“…Define Γ k " P ˚xh k y. We can now use [IRV,Proposition 5.3] to conclude that lim kÑ8 δ Γ k " δ P . Moreover, the manifold H n {Γ k is geometrically finite (see [DP] for a proof when P has rank one, but the same argument applies to Γ k ).…”