Symbolic Dynamics, Smooth Dynamics, and Applications

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The proposed project has several related components: 1) Weplan to continue our study of the Schelling segregationmodel as a dynamical system. This model, which first arosein economics, is related to a number of lattice models instatistical physics like the lattice gas, but more difficultdue to the inherent non-local nature of site coupling; 2) Weplan to study the "rigidity" of periodic point invariantsfor symbolic and hyperbolic dynamical systems. Thesetopological invariants include, for a Holder continuousfunction f, the unmarked periodic orbit spectrum, the betafunction P(-s f), and the zeta function. These invariantsare fundamental objects of study in dynamics and statisticalphysics, but the information about the function f theycapture is subtle and poorly understood; 3) We plan tocontinue our investigation into the distribution of valuesof fundamental quantities in ergodic theory (e.g. Lyapunovexponents, local entropy, and Birkhoff averages) and thefine structure of the corresponding phase spacedecomposition.The proposed project has several related components: 1) Weplan to continue our study of the Schelling segregationmodel as a dynamical system. This model, which was firstproposed by the eminent economist Thomas Schelling, isrelated to a number of lattice models in statistical physicslike the lattice gas, but more difficult due to the inherentnon-local nature of site coupling; 2) Pressure is afundamental object of study in statistical physics, but evenin highly idealized systems, the information about thesystem it captures is subtle and poorly understood. We planto study whether certain systems are completely identifiedby their pressure. These problems have striking similaritiesto fascinating questions which Kac adroitly summarized withthe question "Can you hear the shape of a drum?"; (3) Forergodic systems, the time average of a function along almostevery orbit equals the spatial average. Only very rarely canalmost every orbit be replaced by every orbit. We plan tostudy the fine structure and dimension of the exceptionalset whose time average does not coincide with the spatialaverage