Discrete event dynamical systems with application to the geodesic flow on hyperbolic surfaces

Author(s)

Daniel Pedro Bezerra Chaves

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Faculdade de Engenharia Elétrica e de Computação
• Defense date: 2011-05-12

Examining board members

Reginaldo Palazzo Júnior; Carlos Eduardo Camara; Cecilio Jose Lins Pimentel; Henrique Lazari; Jose Roberto Rios Leite

• Advisor: Reginaldo Palazzo Júnior

Abstract

In this work we present methods for a combinatorial description of the geodesic flow on a hyperbolic compact surface, with the intent of identifying how the topological parameters of the surface may be associated with discrete sequences. This approach allows to conjugate the topological and combinatorial properties of a surface with concepts of information theory and coding. We determine the intrinsic topological property of complete and time-invariant discrete dynamical systems whose trajectories are bi-infinite sequences over a finite alphabet. The same structure generated by the geodesic flow coding methods, that we call shift space. We show how a shift space can be completely characterized by the irreducible forbidden set and locally by the constraint sets, and how both can be obtained through partial order relations. As consequence of these results, some constructions to represent the shift spaces are proposed. Methods for coding source sequences on hyperbolic surfaces are proposed, based on T-piecewise and common-sets relations that exist between these methods. We conclude by specifying a construction procedure for presentations of arithmetic codes that is related with the topological and combinatorial properties of the hyperbolic surface (AU)