Shannon-McMillan-Breiman theorem along almost geodesics in negatively curved groups

Abstract

Consider a non-elementary Gromov-hyperbolic group $\Gamma$ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on $(X,\mu)$. We construct special increasing sequences of finite subsets $F_n(y)\subset \Gamma$, with $(Y,\nu)$ a suitable probability space, with the following properties: given any countable partition $\mathcal{P}$ of $X$ of finite Shannon entropy, the refined partitions $\bigvee_{\gamma\in F_n(y)}\gamma \mathcal{P}$ have normalized information functions which converge to a constant limit, for $\mu$-almost every $x\in X$ and $\nu$-almost every $y\in Y$; the sets $\mathcal{F}_n(y)$ constitute almost-geodesic segments, and $\bigcup_{n\in \mathbb{N}} F_n(y)$ is a one-sided almost geodesic with limit point $F^+(y)\in \partial \Gamma$, starting at a fixed bounded distance from the identity, for almost every $y\in Y$; the distribution of the limit point $F^+(y)$ belongs to the Patterson-Sullivan measure class on $\partial \Gamma$ associated with the invariant hyperbolic metric. The main result of the present paper amounts therefore to a Shannon-McMillan-Breiman theorem along almost geodesic segments in any p.m.p. action of $\Gamma$ as above. For several important classes of examples we analyze, the construction of $F_n(y)$ is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all $\Gamma$-generating partitions of $X$. Using an important inequality due to B. Seward, we deduce that it is equal to the Rokhlin entropy $\frak{h}^{\text{Rok}}$ of the $\Gamma$-action on $(X,\mu)$, provided that the action is free.