Symbolic Dynamics, Smooth Dynamics, and Applications

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

The proposed project has several related components: 1) We plan to continue our study of the Schelling segregation model as a dynamical system. This model, which first arose in economics, is related to a number of lattice models in statistical physics like the lattice gas, but more difficult due to the inherent non-local nature of site coupling; 2) We plan to study the "rigidity" of periodic point invariants for symbolic and hyperbolic dynamical systems. These topological invariants include, for a Holder continuous function f, the unmarked periodic orbit spectrum, the beta function P(-s f), and the zeta function. These invariants are fundamental objects of study in dynamics and statistical physics, but the information about the function f they capture is subtle and poorly understood; 3) We plan to continue our investigation into the distribution of values of fundamental quantities in ergodic theory (e.g. Lyapunov exponents, local entropy, and Birkhoff averages) and the fine structure of the corresponding phase space decomposition. The proposed project has several related components: 1) We plan to continue our study of the Schelling segregation model as a dynamical system. This model, which was first proposed by the eminent economist Thomas Schelling, is related to a number of lattice models in statistical physics like the lattice gas, but more difficult due to the inherent non-local nature of site coupling; 2) Pressure is a fundamental object of study in statistical physics, but even in highly idealized systems, the information about the system it captures is subtle and poorly understood. We plan to study whether certain systems are completely identified by their pressure. These problems have striking similarities to fascinating questions which Kac adroitly summarized with the question "Can you hear the shape of a drum?"; (3) For ergodic systems, the time average of a function along almost every orbit equals the spatial average. Only very rarely can almost every orbit be replaced by every orbit. We plan to study the fine structure and dimension of the exceptional set whose time average does not coincide with the spatial average