Ergodic Theory of Smooth One-Dimensional Maps

动力系统作为天体力学、气象学、流体对流、经济、社会科学和人口动力学中重要现象的模型自然出现。即使是一维的动力系统，这个研究项目的重点，也可以表现出极其丰富和复杂的特性，这对理解来说是一个挑战。本研究项目的目标是使用概率论、遍历理论和统计力学的思想和技术来研究一维动力系统，与这些系统相关的各种对象的参数依赖性，如物理度量和几何吉布斯状态，以及分形性质的集合，如Julia集合和Mandelbrot集合。本课题主要研究一维大偏差原理的普适性；Pesin关于典型logistic映射不存在相变的猜想线性响应理论的Ruelle猜想以及一些具有分形性质的集合的精细几何，比如Julia集合和Mandelbrot集合。研究生和本科生将参与该项目。

Dynamical systems appear naturally as models of important phenomena in celestial mechanics, meteorology, fluid convection, economy, social sciences, and population dynamics. Even dynamical systems in one dimension, the focus of this research project, can exhibit extremely rich and intricate properties that represent a challenge to understanding. The goal of this research project is to use ideas and techniques from probability, ergodic theory, and statistical mechanics to study one-dimensional dynamical systems, the dependence on parameters of various objects associated to these systems, such as physical measures and geometric Gibbs states, and sets of fractal nature like Julia sets and the Mandelbrot set.This project focuses on investigating the universality of the large deviation principle in dimension one; Pesin's conjecture on the absence of phase transitions for typical logistic maps; Ruelle's conjecture on linear response theory; and the fine geometry of some sets of fractal nature such as Julia sets and the Mandelbrot set. Graduate and undergraduate students will be involved in the project.