# Ergodic Theory of Parabolic Horseshoes

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## Abstract

In this paper we develop the ergodic theory for a horseshoe map *f* which is uniformly hyperbolic, except at one parabolic fixed point *ω* and possibly also on *W* ^{ s }(*ω*). We call *f* a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of *σ*-finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of *σ*-finite equilibrium states (each classical probability equilibrium state is a *σ*-finite equilibrium state). We then study the unstable pressure function \({t \mapsto P(-t \log |Df| E^u|)}\), the corresponding finite and *σ*-finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic *σ*-finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two *σ*-finite ergodic conservative equilibrium states for the potential –*t* ^{ u }log |*Df*|*E* ^{ u }| (where *t* ^{ u } denotes the unstable dimension), one of which is the Dirac *δ*-measure supported at the parabolic fixed point and the other being non-atomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension.

## Keywords

Ergodic Theory Unstable Manifold Hausdorff Dimension Maximal Dimension Iterate Function System## Preview

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