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The Categorical Basis of Dynamical Entropy

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Abstract

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system—which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.

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The research is not dependent on any data, experimental or otherwise. Please contact the author Suddhasattwa Das for any further information.

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  1. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA

    Suddhasattwa Das

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The sole author Suddhasattwa Das was responsible for all the research and in the preparation of the manuscript.

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Communicated by Nicola Gambino.

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Das, S. The Categorical Basis of Dynamical Entropy. Appl Categor Struct 32, 8 (2024). https://doi.org/10.1007/s10485-024-09763-2

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