Exponential decay of Lebesgue numbers
Let \(f: X \to X\) be a continuous map acting on a compact metric space. Denote by \(\delta(\mathcal{U})\) the Lebesgue number of a cover \(\mathcal{U}\), i.e., the largest number such that any ball of radius less than \(\delta(\mathcal{U})\) is contained in some element of \(\mathcal{U}\). Define \(h^-_L(f)=\sup h_L^-(f,\mathcal{U})\) and \(h^+_L(f)=\sup h_L^+(f,\mathcal{U})\), where NEWLINE\[NEWLINE h_L^+(f,\mathcal{U})=-\limsup \frac{1}{n}\log \delta_n (f,\mathcal{U}),\quad h_L^-(f,\mathcal{U})=-\liminf \frac{1}{n}\log \delta_n (f,\mathcal{U}) NEWLINE\]NEWLINE and \(\delta_n(\mathcal{U})=\min_{0\leq k <n}\delta(f^{-k}(\mathcal{U}))\).NEWLINENEWLINEThe paper contains many interesting results on relations between Lebesgue numbers, entropy and dimension. For example, it is proved that NEWLINE\[NEWLINE \dim _B(X) h_L^-(f)\geq h(f)\text{ and }\dim _H(X) h_L^+(f)\geq h(f) NEWLINE\]NEWLINE where \(h(f)\) is topological entropy, and \(\dim _B\), \(\dim _H\) are, respectively, the upper box dimension and the Hausdorff dimension. It is also proved that if \(f\) is Lipshitz with constant \(L(f)>0\) then \(h_L^+(f,\mathcal{U})\leq \max \{\log L(f),0\}\) for every open cover \(\mathcal{U}\).NEWLINENEWLINEThe paper ends with a few examples showing how developed results apply in practice.
scientific article; zbMATH DE number 6072022