Equilibrium States for Center Isometries

Abstract

We develop a geometric method to establish existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular the entropy maximizing measure and the SRB measure. It is also given a characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

Figures (1)

• Figure 1: Comparison between $P^{cu}(x,\epsilon),P^{cu}(y,\epsilon)$ for $y\in W^s(x)$. The stable holonomy sends each center plaque $W^c(x',\epsilon)\in P^{cu}(x,\epsilon)$ to a center plaque $W^c(y',\epsilon)\in P^{cu}(y,\epsilon)$, but the image of unstable plaques in $P^{cu}(x,\epsilon)$ are only Hölder submanifolds in $P^{cu}(y,\epsilon)$. Nevertheless, $\mathrm{hol}^s_{x,y}(W^u(x,\epsilon))\to W^u(x,\epsilon)$ in the $\mathcal{C}^0$ topology as $y\to x$

Theorems & Definitions (154)

• Theorem 1: Variational Principle
• Definition 1
• Theorem 2
• Remark 1
• Definition 2
• Remark 2
• Lemma \oldthetheorem
• proof
• Definition 3
• Theorem \oldthetheorem