Shannon–McMillan–Breiman Theorem Along Almost Geodesics in Negatively Curved Groups

Consider a non-elementary Gromov-hyperbolic group Γ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on (X,μ ) . We construct special increasing sequences of finite subsets F_n(y)⊂Γ , with (Y,ν ) a suitable probability space, with the following properties. Given any countable partition 𝒫 of X of finite Shannon entropy, the refined partitions ⋁ _γ∈ F_n(y)γ𝒫 have normalized information functions which converge to a constant limit, for μ -almost every x∈ X and ν -almost every y∈ Y . The sets ℱ_n(y) constitute almost-geodesic segments, and ⋃ _n∈ℕ F_n(y) is a one-sided almost geodesic with limit point F^+(y)∈∂Γ , starting at a fixed bounded distance from the identity, for almost every y∈ Y . The distribution of the limit point F^+(y) belongs to the Patterson–Sullivan measure class on ∂Γ 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 Γ 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 Γ -generating partitions of X . Using an important inequality due to Seward (Weak containment and Rokhlin entropy, arxiv:1602.06680 , 2016), we deduce that it is equal to the Rokhlin entropy 𝔥^Rok of the Γ -action on (X,μ ) defined in Seward (Invent Math 215:265–310, 2019), provided that the action is free. Remarkably, this property holds for every choice of invariant hyperbolic metric, every choice of suitable auxiliary space (Y,ν ) and every choice of special family F_n(y) as above. In particular, for every ϵ > 0 , there is a generating partition 𝒫_ϵ , such that for almost every y∈ Y , the partition refined using the sets F_n(y) has most of its atoms of roughly constant measure, comparable to exp (-n𝔥^Rok±ϵ ) . This describes an approximation to the Rokhlin entropy in geometric and dynamical terms, for actions of word-hyperbolic groups.