论文标题
从热力学和光谱相变到多重分析
From thermodynamic and spectral phase transitions to multifractal analysis
论文作者
论文摘要
众所周知，所有均匀扩展或双曲动力学都没有相对于Hölder持续电势的相变。 In \cite{BC21}, is proved that for all transitive $C^{1+α}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transfer operator $\mathcal{L}_{f,-t\log|Df|}$ acting on the space of Hölder continuous functions, has the spectral gap property for所有$ t <t_0 $都没有所有$ t \ geq t_0 $的光谱间隙属性。中央Lyapunov频谱的多重分析。
It is known that all uniformly expanding or hyperbolic dynamics have no phase transition with respect to Hölder continuous potentials. In \cite{BC21}, is proved that for all transitive $C^{1+α}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transfer operator $\mathcal{L}_{f,-t\log|Df|}$ acting on the space of Hölder continuous functions, has the spectral gap property for all $t<t_0$ and does not have the spectral gap property for all $t\geq t_0$. Our goal is to prove that the results of thermodynamical and spectral phase transitions imply a multifractal analysis for the Lyapunov spectrum. In particular, we exhibit a class of partially hyperbolic endomorphisms that admit thermodynamical and spectral phase transitions with respect to the geometric potential, and we describe the multifractal analysis of your central Lyapunov spectrum.
