Genericity of continuous maps with positive metric mean dimension (Q2242605)

This paper is primarily concerned with the metric mean dimension of a dynamical system, a concept that is related to topological entropy. The definition of the lower metric mean dimension, \(\underline{\mathrm{mdim}}_M(X,d,\phi)\), in this paper coincides with the definition of metric mean dimension in [\textit{E. Lindenstrauss} and \textit{B. Weiss}, Isr. J. Math. 115, 1--24 (2000; Zbl 0978.54026)]. The upper metric mean dimension, \(\overline{\mathrm{mdim}}_M(X,d,\phi)\), is a slight modification of the lower. In the case that both the upper and lower metric mean dimensions coincide, their shared value is referred to as the metric mean dimension. The first example is a path of topologically conjugate maps on the interval \([0,1]\) with the Euclidean metric so that no two maps in this path agree on their metric mean dimensions. This paper also establishes that, for two compact metric spaces \((X,d)\) and \((Y,d')\) with continuous \(\phi : X \to X\) and \(\psi : Y \to Y\), there holds: \begin{itemize} \item \(\overline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi) \leq \overline{\mathrm{mdim}}_M(X, d, \phi) + \overline{\mathrm{mdim}}_M(Y , d' , \psi)\); \item \(\overline{\mathrm{mdim}}_M(X, d, \phi) + \underline{\mathrm{mdim}}_M(Y , d' , \psi) \leq \overline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi)\); \item \(\underline{\mathrm{mdim}}_M(X, d, \phi) + \underline{\mathrm{mdim}}_M(Y , d' , \psi) \leq \underline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi)\); \item \(\underline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi) \leq \underline{\mathrm{mdim}}_M(X, d, \phi) + \overline{\mathrm{mdim}}_M(Y , d' , \psi)\). \end{itemize} Consequently, if either \(\underline{\mathrm{mdim}}_M(X, d, \phi) = \overline{\mathrm{mdim}}_M(X, d, \phi)\) or \(\underline{\mathrm{mdim}}_M(Y, d', \psi) = \overline{\mathrm{mdim}}_M(Y, d', \psi)\), then \[ \overline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi) = \overline{\mathrm{mdim}}_M(X, d, \phi) + \overline{\mathrm{mdim}}_M(Y , d' , \psi) \] and \[ \underline{\mathrm{mdim}}_M(X \times Y , d \times d' , \phi \times \psi) = \underline{\mathrm{mdim}}_M(X, d, \phi) + \underline{\mathrm{mdim}}_M(Y , d' , \psi). \] By considering the box dimension and the left shift map, the author shows that the above inequalities can be strict. The rest of the paper is focused on the density of maps with positive metric mean dimension. It is shown that the set of continuous maps \(N \to N\), where \(N\) is a compact Riemannian manifold, with metric mean dimension \(a \in [0, \mathrm{dim}(N)]\) is dense in the space of continuous functions \(N \to N\); furthermore, the set of maps with upper metric mean dimension equal to \(\mathrm{dim}(N)\) is residual. The primary method used to establish this result is perturbations on small neighborhoods of the orbit of a periodic point. As a consequence, it is noticed that the metric mean dimension function on the space of continuous functions \(N \to N\) is not continuous anywhere. Lower and upper semi-continuity are similarly addressed. By investigating countably infinite products of finite spaces, similar results to those above regarding the density of maps with positive metric mean dimension and the discontinuity of the metric mean dimension function are established for perfect, compact, metrizable, zero-dimensional spaces.