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

Rhiannon Dougall, Richard Sharp

Research output: Contribution to journalArticle (Academic Journal)

10 Downloads (Pure)

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 $X^{\mathrm{ab}}$. In this way, we obtain an equivalence for the Gurevi\v{c} entropy: $h(X)=h(X^{\mathrm{ab}})$ 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.
Original languageEnglish
Number of pages39
JournalInventiones Mathematicae
Early online date26 Aug 2020
DOIs
Publication statusE-pub ahead of print - 26 Aug 2020

Fingerprint Dive into the research topics of 'Anosov flows, growth rates on covers and group extensions of subshifts'. Together they form a unique fingerprint.

  • Cite this