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Statistical properties of weak Gibbs measures for complex systems with nonhyperbolic periodic orbits

非双曲周期轨道复杂系统弱吉布斯测度的统计特性

基本信息

批准号：
13640133
负责人：
YURI Michiko
金额：
$ 1.92万
依托单位：
Sapporo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640133/
关键词：
Equilibrium state Weak Gibbs measure Variational principle Intermittency Indifferent periodic point Conformal measure Multifractal formalism Zeta function Variational Principle Intermittency indifferent periodic point local product s

项目摘要

The purpose of this project is to construct mathematical models which show phase transition, failure of the Gibbs property and slow decay of correlations, that can be observed for many complex systems exhibiting Intermittency. For this purpose, we establish Thermodynamic formalism for non-Holder potentials in nonhyperbolic situation. More specifically, for countable to one transitive Markov systems we construct conformal measures ν which are weak Gibbs measures for potentials φ of weak bounded variation, and show the existence of equilibrium states μ for φ equivalent to the weak Gibbs measures ν ([1]). We see that generalized indifferent periodic orbits cause phase transition, non-Gibbsianness and force the decay of correlations to be slow ([1], [4]). Applying these results allow us to establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation ([3]). Furthermore, we establish a version of the local product structure (weak local product structure) for the invertible extension of ergodic weak Gibbs measures μ ([2]). About the decay of correlations of weak Gibbs equilibrium measures, polynomial upper bounds can be obtained for a class of functions which contains all piecewise Lipschitz functions ([4]), by clarifying the speed of uniform convergence of iterated transfer operators on compact sets excluding indifferent periodic points.

该项目的目的是构建数学模型，显示相变，吉布斯性质的失败和相关性的缓慢衰减，可以观察到许多复杂的系统表现出间歇性。为此，我们建立了非双曲情形下非Holder势的热力学形式。更具体地说，对于可数到一传递的马氏系统，我们构造了弱有界变差势φ的弱Gibbs测度共形测度ν，并证明了φ的平衡态μ等价于弱Gibbs测度ν（[1]）.我们看到，广义无关周期轨道引起相变，非吉布斯性，并迫使关联衰减变慢（[1]，[4]）。应用这些结果使我们能够建立与弱有界变差势相关的弱吉布斯测度的多重分形形式（[3]）。进一步，我们建立了遍历弱Gibbs测度μ（[2]）的可逆扩张的局部积结构（弱局部积结构）.关于弱Gibbs平衡测度的相关性的衰减，通过阐明迭代转移算子在不含无关周期点的紧集上的一致收敛速度，得到了一类包含所有分段Lipschitz函数的函数的多项式上界（[4]）.