Ruelle's inequality in negative curvature

Abstract

In this paper we study different notions of entropy for measurepreserving 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.

Central limit theorem of Brownian motions in pinched negative curvature

“…Thus F su is Hölder continuous and the Liouville measure on T 1 M is the equilibrium state for F su . The existence with an assumption on the pressure of F su is proved in Chapter 7 of [27] and [32] proves that the assumption is true in our case. The measure class determined by the Patterson-Sullivan density is called the Lebesgue class.…”

A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures

“…Remark 2.1. It follows from Theorem 2.10 in [19] that if X is a complete (non-compact) Riemannian manifold and µ ∈ M e (f ), then (15) is also valid.…”

“…Now, if µ ∈ M e (f ), it follows from Lemma 2.8 in [19] that h µ (f, x) = µ-ess inf h µ (T, y) is valid for µ-a.e. x, and then, by Theorem 2.9 in [19], that h µ (f, x) ≥ h µ (T ) is also valid for µ-a.e.…”

Generic properties of invariant measures of full-shift systems over perfect separable metric spaces

“…One also has, by Lemma 2.8 in [32], that h µ (T, x)dµ(x) = µ-ess inf h µ (T, x), and then, by Theorem 2.9 in [32], that h µ (T, x)dµ(x) ≥ h µ (T ). Thus, by inequality (7), one gets for µ-a.e.…”