Anosov flows, growth rates on covers and group extensions of subshifts

Abstract

The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem of counting periodic orbits in an amenable cover X to counting in a maximal abelian subcover X ab. In this way, we obtain an equivalence for the Gurevič entropy: h(X) = h(X ab) if and only if the covering group is amenable. In addition, when we project the periodic orbits for amenable covers X to the compact factor M, they equidistribute with respect to a natural equilibrium measure-in the case of the geodesic flow, the measure of maximal entropy.

Drift and Matrix coefficients for discrete group extensions of countable Markov shifts

“…Assume now that Σ is compact. For amenable groups the work of [9] shows that the Gurevič pressure P (log R, T s ) is equal P (log R + ψ) for a unique real onedimensional representation π : G → R and ψ = π • s. (In other words: let ψab be the composition of…”

“…Then there is a unique ξ ∈ R that determines ψ(x) = ξ, ψab (x) R d .) We use similar ideas to [16] describing drift for abelian extensions; and the same ideas behind the equidistribution result of [9].…”

“…(This can be encoded to a weak symmetry hypothesis for a symbolic coding.) Dougall and Sharp [9] study the entropy of an Anosov flow in a covering manifold -in the context of Anosov flows one can already have a drop in entropy for the homology cover [16]. A particular case of their technique characterises the radius of convergence of a finitely supported random walk on an amenable group G as being equal to the radius of convergence for the abelianisation of G, with the restriction that the semigroup G + p generated by the support of the probability p is in fact the whole of G. (In particular there is no symmetry requirement.)…”

“…This was proved in the framework of transitive group extensions of subshifts of finite type (and in that context G + p = G implies transitivity of the group extension). The characterisation of the radius of convergence for an amenable group had not been seen before the work of [9]. We expand on the technique of [9] to give Theorem 3.16, and compare this with a ratio limit theorem (see subsection 2.2).…”

“…The characterisation of the radius of convergence for an amenable group had not been seen before the work of [9]. We expand on the technique of [9] to give Theorem 3.16, and compare this with a ratio limit theorem (see subsection 2.2).…”