Anqie entropy and arithmetic compactification of natural numbers

Fei Wei Yau Mathematical Sciences Center, Tsinghua University

Dynamical Systems Functional Analysis mathscidoc:2205.11012

Banach Journal of Mathematical Analysis, 16, (1), 11, 2021.11
To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.
Anqie entropy, arithmetic compactification, C*-algebra, $K$-groups, totally disconnected
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  title={Anqie entropy and arithmetic compactification of natural numbers},
  author={Fei Wei},
  booktitle={Banach Journal of Mathematical Analysis},
Fei Wei. Anqie entropy and arithmetic compactification of natural numbers. 2021. Vol. 16. In Banach Journal of Mathematical Analysis. pp.11.
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