Anqie entropy and arithmetic compactification of natural numbers (Q2062189)

The notion of entropy is an important tool to measure some uncertainty of a dynamical system. The classical measure-theoretic entropy in ergodic theory was first introduced by \textit{A. N. Kolmogorov} [Dokl. Akad. Nauk SSSR 119, 861--864 (1958; Zbl 0083.10602)] and \textit{Ya. G. Sinai} [Dokl. Akad. Nauk SSSR 124, 768--771 (1959; Zbl 0086.10102)] while the classical topological entropy in topological dynamics appeared in the work by \textit{R. L. Adler} et al. [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)]. In this paper, the author introduces the notion of ``anqie entropy'' of arithmetic functions to study some arithmetic structures of natural numbers. Let \(\mathcal{A}\) be a unital \(C^*\)-subalgebra of \(l^{\infty}(\mathbb{N})\). We call \(\mathcal{A}\) an anqie of \(\mathbb{N}\) if \(\mathcal{A}\) is invariant under the map \(\sigma_A\) on \(l^{\infty}(\mathbb{N})\), where \(\sigma_A\) is defined by \((\sigma_Af)(n)=f(n+1)\) for any \(f \in l^{\infty}(\mathbb{N})\) and \(n \in \mathbb{N}\). From the viewpoint of dynamical systems, an anqie of \(\mathbb{N}\) is a topological dynamical system associated with the additive structure of \(\mathbb{N}\). The anqie entropy of an anqie \(\mathcal{A}\) is defined to be the topological entropy of the additive map \(A\) on \(X\). It is proved that the set of all arithmetic functions with zero anqie entropy \(\mathcal{E}_0(\mathbb{N})\) forms a \(C^*\)-algebra. The maximal ideal space of \(\mathcal{E}_0(\mathbb{N})\) defines the arithmetic compactification of natural numbers, which is denoted by \(E_0(\mathbb{N})\). From the topological viewpoint, the arithmetic compactification is the maximal zero entropy topological factor of the Stone-Čech compactification of \(\mathbb{N}\). The arithmetic and Stone-Čech compactification of \(\mathbb{N}\) are both uncountable and unmetrizable. Furthermore, the Stone-Čech compactification is extremely disconnected and totally disconnected, while for the arithmetic compactification, the author obtains that it is totally disconnected but not extremely disconnected. The operator \(K\)-theory provides an useful tool to learn about the structure of \(C^*\)-algebras. \(K\)-groups can be regarded as invariants to distinguish two \(C^*\)-algebras. For \(\mathcal{E}_0(\mathbb{N})\), the group \(K_0(\mathcal{E}_0(\mathbb{N}))\) is homeomorphic to the additive group \(\{f \in \mathcal{E}_0(\mathbb{N}): f(\mathbb{N}) \subseteq \mathbb{Z}\}\), and the group \(K_1(\mathcal{E}_0(\mathbb{N}))\) is trivial. As an application, the author obtains an approximation result for topological dynamical systems, that is, any topological dynamical system with topological entropy \(\lambda\) can be approximated by symbolic dynamical systems with entropy less than or equal to \(\lambda\).