Three different entropies, variational principle and the degree formula.

My inaugural blog post, hurray!! This post is based on a mandatory assignment in a course in complex dynamics held at the university of Oslo 2018. For sources we mainly the book of Walters [1] and handouts. Some of the more basic results will, due to time constraints, be left unproven, but can all be found in [1].

– Introduction –

The notion of entropy, first formulated in the context of thermodynamics, is a quantity meant to capture the complexity/chaoticness of the state of a system. It is maybe not surprising that such a broadly defined notion spread like aids throughout different subfields of applied mathematics and information theory. Here, three different definitions of entropy are introduced, two of which turn out to be equivalent when working on compact metric spaces, and all three are related, on compact metric spaces, by the so called Variational Principle.
Lastly, the theorem of Misiurewicz, Przytycki and Gromov, which relates the entropy of a holomorphic map on the $ n$-sphere to its degree was proven.

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