r-中和Pfister-Sullivan熵的变分原理

中文摘要 AI 辅助

本文研究r-中和Pfister-Sullivan熵，该熵通过测量经验测度接近指定不变测度的轨道段的复杂性来定义。我们主要得到以下三个结果：(i) 我们反驳了C. Dong和Q. Qiao提出的一个变分猜想[《关于r-中和熵：熵公式与达到上确界的测度的存在性》，Comm. Math. Phys. 406 (2025)，文章编号74]。(ii) 我们得到了上r-中和拓扑熵和下r-中和拓扑熵关于概率测度的变分公式，还通过PS局部化引入了新的测度论量，并建立了上r-中和拓扑熵的一些变分原理。(iii) 我们在不同光滑性假设下得到了r-中和PS熵的上界，以及对于C^{1+α}微分同胚的下界，这些下界用熵、维数和李雅普诺夫指数表示；对于C^∞微分同胚和C^1 Anosov微分同胚，对每个遍历测度μ，r-中和PS熵等于h_μ(f)+r dim M；我们还建立了混合曲面双曲基本集上的精确公式，并证明对于C^2曲面微分同胚，全流形测度公式可能不成立。

英文摘要

In this paper, we study $r$-neutralized Pfister--Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On $r$-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower $r$-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper $r$-neutralized topological entropy (iii) We obtain upper bounds for $r$-neutralized PS entropy under different smoothness assumptions and lower bounds for $C^{1+α}$ diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For $C^\infty$ diffeomorphisms and $C^{1}$ Anosov diffeomorphisms, the $r$-neutralized PS entropies equal $h_μ(f)+r\dim M$ for every ergodic measure $μ$. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for $C^2$ surface diffeomorphisms.

发表机构

• Nanjing Normal University（南京师范大学）

机构由 AI 辅助整理，请以论文原文为准。