非一致双曲流的符号动力学
Symbolic dynamics for non-uniformly hyperbolic flows
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中文总结 AI 辅助
本文针对含不动点的任意维度非一致双曲流构建符号动力学，通过修改黎曼度量处理向量场奇点，证明三维正拓扑熵C^∞流的极大熵遍历测度有限，并给出周期轨道数量及伯努利性质的相关应用。
中文摘要 AI 辅助
我们针对任意维度、可能包含不动点的非一致双曲流构建符号动力学。更确切地说，对任意χ>0，我们编码得到一个集合，该集合对每个χ-双曲不变概率测度具有全测度，且该测度对不动点集合赋予零质量。作为主要应用，我们证明闭流形上具有正拓扑熵的三维C^∞流具有有限多个极大熵遍历测度。对任意维度的流，我们还给出了周期轨道数量的应用，以及Hölder连续势的平衡态的伯努利性质的应用。本文的主要技术结果是一种通过修改黎曼度量处理向量场奇点的新方法，该技术类似于爆破，可将许多非奇异向量场的结果直接应用于带奇点的向量场。
英文摘要
We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $χ>0$, we code a set which has full measure for every $χ$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of Hölder continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.
