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arXiv 1902.01690math.DS
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不含控制分裂的保守$C^1$ -纯同态的拓扑压力

Topological pressure for conservative $C^1$-diffeomorphisms with no dominated splitting

Xueming Hui

备注：
30 pages. This article is a generalization of arXiv:1606.01765. In this version, some typos are corrected and some proofs are modified to give stronger results
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摘要：

证明了$C^1$ -一般保守超同态的拓扑压的三个计算公式，并证明了拓扑压关于这些超同态的连续性。我们证明了对于这些一般的超同态，不存在具有正测度理论熵的平衡态。特别地，对于双曲势，不存在平衡态。对于无支配分裂的紧致曲面上的$C^1$类属保守同胚和$ϕ_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$，我们证明了存在熵为零的平衡态，并且对于族$\lbrace t ϕ_m(x)\rbrace_{t\geq 0}$存在一个过渡点$t_0$，使得$ t \in [0, t_0)$不存在平衡态，而$t \in [t_0,+\infty)$存在一个平衡态.

英文摘要：

We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there is no equilibrium states. For $C^1$ generic conservative diffeomorphism on compact surfaces with no dominated splitting and $ϕ_m(x):=-\frac{1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with zero entropy and there exists a transition point $t_0$ for the family $\lbrace t ϕ_m(x)\rbrace_{t\geq 0}$, such that there is no equilibrium states for $ t \in [0, t_0)$ and there is an equilibrium state for $t \in [t_0,+\infty)$.

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