Topological Indices in Discreat Dynamical Systems and Application

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Shahidi Shadkam, Sheida | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 44375 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Razvan, Mohammad Reza
Abstract:
In order to generalize the Morse Index for fixed points of a flow, Conley introduced an invariant for isolated invariant sets under a flow which is known as Conley Index.However a similar generalization in discrete dynamical systems is more difficult. Category theory, shape theory and algebraic topology have been used in the generalization. In this thesis we propose a construction of Conley index for discrete dynamical systems, which goes along Conley lines as long as it was possible, but certain modifications has been done for convenience. After introducing the index, we will consider some applications: the generalized Morse inequalities and the Morse decomposition. At the end we will express the relation between the discrete Conley index and existence of periodic orbit for a continuous dynamical system
Keywords:
Conley Index ; Discrete Dynamical System ; Morse Decomposition ; Poincare Polynomials ; Poincare Section

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