Ergodic Theory of Smooth One-Dimensional Maps

University Of Rochester, Rochester NY

Investigators

Abstract

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.