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Consider a non-elementary Gromov-hyperbolic group $\Gamma$ with a suitableinvariant hyperbolic metric, and an ergodic probability measure preserving(p.m.p.) action on $(X,\mu)$. We construct special increasing sequences offinite subsets $F_n(y)\subset \Gamma$, with $(Y,\nu)$ a suitable probabilityspace, 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 informationfunctions which converge to a constant limit, for $\mu$-almost every $x\in X$and $\nu$-almost every $y\in Y$; the sets $\mathcal{F}_n(y)$ constitutealmost-geodesic segments, and $\bigcup_{n\in \mathbb{N}} F_n(y)$ is a one-sidedalmost geodesic with limit point $F^+(y)\in \partial \Gamma$, starting at afixed bounded distance from the identity, for almost every $y\in Y$; thedistribution of the limit point $F^+(y)$ belongs to the Patterson-Sullivanmeasure class on $\partial \Gamma$ associated with the invariant hyperbolicmetric. The main result of the present paper amounts therefore to aShannon-McMillan-Breiman theorem along almost geodesic segments in any p.m.p.action of $\Gamma$ as above. For several important classes of examples weanalyze, the construction of $F_n(y)$ is purely geometric and explicit.Furthermore, consider the infimum of the limits of the normalized informationfunctions, taken over all $\Gamma$-generating partitions of $X$. Using animportant inequality due to B. Seward, we deduce that it is equal to theRokhlin entropy $\frak{h}^{\text{Rok}}$ of the $\Gamma$-action on $(X,\mu)$,provided that the action is free.

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