Anosov flows, growth rates on covers and group extensions of subshifts
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Anosov flows, growth rates on covers and group extensions of subshifts Rhiannon Dougall1,2 · Richard Sharp3
Received: 29 April 2019 / Accepted: 11 August 2020 © The Author(s) 2020
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 Gureviˇc 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.
B Rhiannon Dougall
[email protected] Richard Sharp [email protected]
1
School of Mathematics, University of Bristol, Bristol BS8 1UG, UK
2
Heilbronn Institute for Mathematical Research, Bristol, UK
3
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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R. Dougall, R. Sharp
1 Introduction 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 imposed in [14] and [42], for example. Our main application is to growth rates of periodic orbits for covers of Anosov flows and we obtain generalisations of results previously known for geodesic flows over compact (or even convex co-compact) negatively curved manifolds [14,35,42]. We begin by describing these results. Let M be a compact smooth Riemannian manifold and let φ t : M → M be a transitive Anosov flow. Then φ t has a countable set of periodic orbits P (φ) and, for γ ∈ P (φ), we write l(γ ) for its period. It is well-known that the growth rate of periodic orbits is given by the topological entropy of φ t ; more precisely 1 log #{γ ∈ P (φ) : l(γ ) ≤ T } =: h = h top (φ). T →∞ T lim
Now suppose that X is a regular cover of M with covering group G, i.e. G acts freely and isometrically on X such that M = X/G. Let φ tX : X → X be the lifted flow, which we assume to be transitive. We will be interested in the growth of periodic orbits for φ tX . If G is finite then φ tX is also an Anosov flow and h top (φ X ) = h top (φ). If G is infinite the situation is more interesting. First, note that if φ tX has a periodic orbit γ then the translates of γ by the action of G give infinitely many periodic orbits with the same period, so a naive definition of periodic orbit growth does not make sense. Rather, we follow the approach of [29] and, choosing an open, relatively compact set W ⊂ X , define h(X ) := lim sup T →∞
1 log #{γ ∈ P (φ X ) : l(γ ) ≤ T, γ ∩ W = ∅}. T
As the notation s
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