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 Gurevic 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.

中文翻译：

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Anosov 流、覆盖的增长率和子班次的组扩展

本文的目的是研究双曲动力系统群扩展的增长特性，其中我们不假设扩展满足对称条件，例如，在 Stadlbauer 的对称群扩展的工作和测地线的作者的工作中看到的流动。我们的主要应用是对 Anosov 流的覆盖的周期轨道的增长率：我们将在一个合适的覆盖 $X$ 中计算周期轨道的问题减少到在最大阿贝尔子覆盖 $X^{\mathrm{ab}}$ 中的计数. 通过这种方式，我们获得了 Gurevic 熵的等价：$h(X)=h(X^{\mathrm{ab}})$ 当且仅当覆盖群是适合的。此外，当我们将适用覆盖 $X$ 的周期轨道投影到紧致因子 $M$ 时，它们相对于自然平衡测度是等分布的——在测地线流的情况下，