Julia集上的Lyapunov指数和Birkhoff谱及复方法在薛定谔算子中的应用

在本项目中，申请人将主要研究复动力系统Julia集上的Lyapunov指数谱和Birkhoff谱以及复方法在离散的薛定谔算子动力系统中Period-doubling模型的应用。 第一部分的问题主要是受到了Przytycki等在Lyapunov指数谱和Sarig关于紧曲面微分同胚的Markov分划两项重要的进展的启发。 我们将研究有理函数Julia集上的Lyapunov指数谱和Birkhoff谱的局部Bowen-Ruelle-Manning-Young公式和关于正熵测遍历度的测度意义的Markov分划。此外还将考虑超越函数的广义Lyapunov指数谱和Birkhoff谱。在第二部分, 我们将利用Eremenko猜想的进展的想法、正规族理论、重整化算子、Phase-energy转移映射等复方法研究Period-doubling薛定谔算子模型的谱集与Julia集的联系。

In this research project, we will study the spectrum of Lyapunov exponents and Birkhoff averages for the Julia set in complex dynamical system and the application of complex methods in Period-doubling discrete Schrodinger operator dynamical system. The research problems in the first part were motivated by the recent progress made by Przytycki and his collaborators on the study of the spectrum of Lyapunov exponents and by Sarig's breakthrough on Markov partition for the diffeomorphism on compact Riemann surface. We will study the local version of Bowen-Ruelle-Manning-Young formula for Lyapunov spectrum and Birkhoff spectrum of rational functions, Markov partition under the setting of ergodic measure with positive entropy. And we also the generalized Lyapunov spectrum and Birkhoff spectrum for the transcendental dynamical system. In the second part, We will analyze the relation between the spectrum of Period-doubling Schrodinger operator and Julia set by applying complex methods of the ideas from recent progress in Eremenko conjecture, normal family theory, renormalized operator, phase-energy transition etc.