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Topological dynamics and applications

Abstract

Topological dynamics is the study of the asymptotic behaviour of one parameter (countable or real) family of transformations considering the existence of periodic attractors (limit cycles), Cantor or fractal attractors, invariant measures, dense orbits and Lyapunov functions. In this research proposal, we investigate the topological dynamics of interval transformations with finitely many discontinuities and maps of Rn (respectively, differential equations on Rn$ whose coordinate functions (respectively, scalar equations) are the composition of an one-variable function with a linear transformation of Rn. Such transformations (respectively, equations) model artificial neural networks, interacting random walks on graphs, switched server systems, strange billiards, exterior billiards and probabilistic models of queues (Queueing Theory). This research proposal also includes the study of the topological dynamics of composition operators on L^p. (AU)

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