Abstract For a measure μ preserved by a C 1 + α 0 ( α 0 > 0 ) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑ e S n ϕ ( x ) over the orbits of some periodic points and the free energy h μ ( f ) + ∫ ϕ d μ (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ ˜ such that for every f-invariant measure supported on Λ ˜ , the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ ˜ such that for every f-invariant measure supported on Δ ˜ , the exponential growth rate equals to the topological pressure.

中文翻译：

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周期轨道上的压力和指数速率

摘要 对于由 C 1 + α 0 ( α 0 > 0 ) 微分同胚 f 和流形上的连续函数 ϕ 保持的度量 μ，我们研究了 ∑ e S n ϕ ( x ) 的指数增长率之间的关系一些周期点的轨道和自由能 h μ ( f ) + ∫ ϕ d μ（在某些情况下，它等于测量理论压力）或拓扑压力。当 μ 是遍历双曲线测度时，我们证明指数增长率与自由能（测量理论压力）一致。并且我们还验证了流形为二维时指数增长率和拓扑压力的相等性。然而，对于高维流形，我们显示了指数增长率和拓扑压力之间的不平等。对于遍历双曲线测度 ω，我们还证明存在一个 ω-full 测量集 Λ ˜ 使得对于 Λ ˜ 支持的每个 f 不变测度，指数增长率等于自由能。此外，我们证明还有另一个 ω-full 测量集 Δ ˜ 使得对于 Δ ˜ 上支持的每个 f 不变测度，指数增长率等于拓扑压力。