Symbolic Dynamics, Smooth Dynamics, and Applications

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