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

Consider a non-elementary Gromov-hyperbolic group $$\Gamma $$ Γ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on $$(X,\mu )$$ ( X , μ ) . We construct special increasing sequences of finite subsets $$F_n(y)\subset \Gamma $$ F n ( y ) ⊂ Γ , with $$(Y,\nu )$$ ( Y , ν ) a suitable probability space, with the following properties. Given any countable partition $$\mathcal {P}$$ P of X of finite Shannon entropy, the refined partitions $$\bigvee _{\gamma \in F_n(y)}\gamma \mathcal {P}$$ ⋁ γ ∈ F n ( y ) γ P have normalized information functions which converge to a constant limit, for $$\mu $$ μ -almost every $$x\in X$$ x ∈ X and $$\nu $$ ν -almost every $$y\in Y$$ y ∈ Y . The sets $$\mathcal {F}_n(y)$$ F n ( y ) constitute almost-geodesic segments, and $$\bigcup _{n\in \mathbb {N}} F_n(y)$$ ⋃ n ∈ N F n ( y ) is a one-sided almost geodesic with limit point $$F^+(y)\in \partial \Gamma $$ F + ( y ) ∈ ∂ Γ , starting at a fixed bounded distance from the identity, for almost every $$y\in Y$$ y ∈ Y . The distribution of the limit point $$F^+(y)$$ F + ( y ) belongs to the Patterson–Sullivan measure class on $$\partial \Gamma $$ ∂ 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 $$\Gamma $$ Γ as above. For several important classes of examples we analyze, the construction of $$F_n(y)$$ F n ( y ) is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all $$\Gamma $$ 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 $$\mathfrak {h}^{\text {Rok}}$$ h Rok of the $$\Gamma $$ Γ -action on $$(X,\mu )$$ ( 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 $$(Y,\nu )$$ ( Y , ν ) and every choice of special family $$F_n(y)$$ F n ( y ) as above. In particular, for every $$\epsilon > 0$$ ϵ > 0 , there is a generating partition $$\mathcal {P}_\epsilon $$ P ϵ , such that for almost every $$y\in Y$$ y ∈ Y , the partition refined using the sets $$F_n(y)$$ F n ( y ) has most of its atoms of roughly constant measure, comparable to $$\exp (-n\mathfrak {h}^{\text {Rok}}\pm \epsilon )$$ exp ( - n h Rok ± ϵ ) . This describes an approximation to the Rokhlin entropy in geometric and dynamical terms, for actions of word-hyperbolic groups.