Dynamical Systems and Ergodic Theory

Ranu, Arantha (2021) Dynamical Systems and Ergodic Theory. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Arantha Ranu (16MS132)) 16MS132_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this project I have explored through different types of discrete dynamical systems but it mainly focuses on two particular types of discrete dynamical systems, which are, the dynamics of a holomorphic map on a Riemann surface and the dynamics of an ergodic map on a measure space. In case of Riemann surface, this project aims at studying the domains of chaos and order for the iterations of a given holomorphic map which are called the Julia and the Fatou sets respectively. The structure of the Fatou and the Julia sets are studied in great detail in this project discussing the number of connected components and the behaviour of the orbit of a point in either domain with increasing time. In the second case, which is the dynamics of ergodic maps on a measure space, this project aims at discussing the important ergodic theorems and using these theorems to give the proof of the fact that every expanding dynamical system is conjugate to some subshift of the finite type with the left shift map. v

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