Arithmetic compactifications of natural numbers and its applications

Fei Wei

Published 2018-11-27Version 1

To study the arithmetic structure of natural numbers, we introduce a notion of entropy of arithmetic functions. This entropy possesses properties common both to Shannon's and Kolmogorov's entropies. We show that all zero-entropy arithmetic functions form a C*-algebra. We introduce the arithmetic compactification of natural numbers as the maximal ideal space of this C*-algebra. The arithmetic compactification is totally disconnected, and not extremely disconnected. We also compute the $K_{0}$-group and the $K_{1}$-group of the space of all continuous functions on the arithmetic compactification. As an application, we prove the following result: suppose that $X$ is a compact finite dimensional manifold with metric $d$, and $T$ a continuous map on $X$ with zero topological entropy. Then for any given $x\in X$ and $\varepsilon>0$, there is a map $f$ from natural numbers to $\{T^{n_{1}}x,T^{n_{2}}x,\ldots, T^{n_{k}}x\}$, namely a finite set of points in the orbit of $x$, such that for each $i=1,\ldots,k$, the characteristic function defined on $f^{-1}(\{T^{n_{i}}x\})$ has zero entropy and $\sup_{n}d(T^{n}x, f(n))< \varepsilon$.