### 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.

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 language | English |
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Number of pages | 39 |

Journal | Inventiones Mathematicae |

Early online date | 26 Aug 2020 |

DOIs | |

Publication status | E-pub ahead of print - 26 Aug 2020 |

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## Profiles

## Dr Rhiannon Dougall

- Heilbronn Institute for Mathematical Research
- School of Mathematics - Heilbronn Research Fellow

Person: Academic , Member