Ergodic Theory of Smooth One-Dimensional Maps

动力系统自然地出现在天体力学、气象学、流体对流、经济学、社会科学和人口动力学中作为重要现象的模型。 即使是一维动力系统，这个研究项目的重点，可以表现出非常丰富和复杂的属性，代表了一个挑战的理解。 这个研究项目的目标是利用概率论、遍历理论和统计力学的思想和技术来研究一维动力系统，研究与这些系统相关的各种对象对参数的依赖性，如物理测量和几何吉布斯状态，以及分形性质的集合，如Julia集和Mandelbrot集。本项目的重点是研究大偏差原理在维度一; Pesin的猜想没有相变的典型逻辑地图; Ruelle猜想的线性响应理论;和精细几何的一些集的分形性质，如朱莉娅集和Mandelbrot集。 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.