An Introduction to Smale Spaces and their Homology

Undergraduate Honors Thesis

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/qn59q5462

Abstract

Dynamics is often described as the evolution of spaces in time. This field is vast and often leads us to fascinating areas such as deterministic chaos, fractals, and applications to other fields of mathematics and science. Many dynamical systems are too chaotic or complicated to study their behavior through direct computation so we often use probabilistic or topological tools to understand them. This thesis will be primarily on the chaotic and hyperbolic systems known as Smale Spaces. Loosely speaking, hyperbolic in the context of dynamics is when an open ball expands in one direction and contracts in another. Their expansive/contractive behavior gives us a plethora of useful properties to study their behavior. In this thesis we will give a brief introduction to the main definitions of topological dynamics. Then, we shall follow Smale and Ruelle's work to give a rigorous introduction to Smale spaces and the main theorems that determine their structure. After this, we shall discuss one of the more important examples known as a shift of finite type which has a totally disconnected topology and can be used to construct a continuous, surjective, dynamic preserving map (factor map) onto any Smale space. We will then introduce Putnam's homology theory for Smale spaces. Finally, discuss new methods of computing this invariant using results obtained during an REU at CU Boulder in 2021 and results of Proietti and Yamashita.

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