Maximizing measures in hyperbolic dynamics

双曲动力学测度最大化

• 批准号: EP/E020801/1
• 负责人: Mark Pollicott
• 金额: $ 32.06万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE020801%2F1
• 关键词: Maximizing; measures; hyperbolic; dynamics

In the study of dynamical systems, the long term asymptotic behaviour is often best understood in terms of properties of invariant measures. In the particular setting of Lagrangian flows, Mane and Mather formulated a number of important questions about those measures whose integrals maximized a particular integral. As part of a general theory, one can ask similar questions about quite diverse dynamical systems. In the particular case of hyperbolic (or chaotic) dynamical systems, we can consider a natural class of measures called Gibbs measures, whose study originated in the mathematical theory of Statistical Mechanics, and the work of Ruelle and Sinai. This proposal relates to how maximizing measures for typical functions for these hyperbolic systems can be approximated by these better behaved Gibbs measures. This has important implications for understanding the quite complicated, but important, maximizing measures in terms of much simpler Gibbs measures.

在动力系统的研究中，长期渐近行为通常可以通过不变测度的性质来最好地理解。在拉格朗日流的特定背景下，马内和马瑟提出了一些关于那些积分最大化特定积分的测度的重要问题。作为一般理论的一部分，人们可以对不同的动力系统提出类似的问题。在双曲（或混沌）动力系统的特定情况下，我们可以考虑一类称为吉布斯测度的自然测度，其研究起源于统计力学的数学理论以及 Ruelle 和 Sinai 的工作。该提案涉及如何通过这些性能更好的吉布斯测度来近似这些双曲系统的典型函数的最大化测度。这对于理解相对简单的吉布斯测度而言相当复杂但重要的最大化测度具有重要意义。

项目成果

How Many Inflections are There in the Lyapunov Spectrum?

• DOI: 10.1007/s00220-021-04161-4
• 发表时间: 2020-02
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: O. Jenkinson;M. Pollicott;P. Vytnova

Thermodynamic Formalism - CIRM Jean-Morlet Chair, Fall 2019

热力学形式主义 - CIRM Jean-Morlet 主席，2019 年秋季

• DOI: 10.1007/978-3-030-74863-0_12
• 发表时间: 2021
• 作者: Pollicott M

Two bifurcation sets arising from the beta transformation with a hole at 0

由 0 处有孔的 beta 变换产生的两个分叉集

• DOI: 10.1016/j.indag.2020.03.001
• 发表时间: 2020
• 期刊: Indagationes Mathematicae
• 作者: Baker S

Uniform lower bounds on the dimension of Bernoulli convolutions

伯努利卷积维数的统一下界

• DOI: 10.1016/j.aim.2021.108090
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Kleptsyn V

Dynamics: Topology and Numbers

动力学：拓扑和数字

• DOI: 10.1090/conm/744/14927
• 发表时间: 2020
• 作者: Sharp R