Anosov flows, growth rates on covers and group extensions of subshifts
Anosov flows, growth rates on covers and group extensions of subshifts
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DOI：
10.1007/s00222-020-00994-3
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发表时间：
2019-04
影响因子：
3.1
通讯作者：
R. Dougall;Richard Sharp
中科院分区：
文献类型：
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作者：
R. Dougall;Richard Sharp
本文的目的是研究双曲动力系统群扩张的增长性质，其中我们不假设扩张满足对称条件，例如Stadlbauer关于对称群扩张和作者关于测地流的工作中所见的对称条件。我们的主要应用是覆盖Anosov流的周期轨道的增长率：我们减少了问题的计数周期轨道在一个顺从的coverXto计数在一个最大的阿贝尔subcover。通过这种方式，我们得到了Gureviews熵的等价性：当且仅当覆盖群是顺从的。此外，当我们将顺从覆盖X的周期轨道投影到紧因子M上时，它们相对于自然平衡测度（在测地线流的情况下，最大熵的测度）等分布。
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 coverXto counting in a maximal abelian subcover. In this way, we obtain an equivalence for the Gurevič entropy:if and only if the covering group is amenable. In addition, when we project the periodic orbits for amenable coversXto the compact factorM, they equidistribute with respect to a natural equilibrium measure — in the case of the geodesic flow, the measure of maximal entropy.