Scale function vs topological entropy (Q386845)

The authors prove that the topological entropy of a topological automorphism of a totally disconnected group is an upper bound of the logarithm of the scale function of the group. The scale function of automorphism was introduced by Willis and was calculated for \(p\)-adic Lie groups and linear groups over local fields. The topological entropy defined in the way of \textit{R. L. Adler} et al. [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)], as well as Bowen, is a concept amply known. The problem of having a control on the entropy is an interesting and very well developed problem. Let \(G\) be a totally disconnected locally compact group and let \(\phi:G\to G\) be a topological automorphism, the scale function of \(G\) is defined as NEWLINE\[NEWLINEs_G(\phi)=\min\{[\phi(U):U\cap\phi(U)] : U\in\mathcal B(G)\},NEWLINE\]NEWLINE where \([H_1 : H_2 ]\) means the index of \(H_1\) in \(H_2\), and \(\mathcal B(G)\) is the set of open compact subgroups of \(G\): One of the authors has obtained a formula for the topological entropy of \(\phi\) in terms of the index of subgroups, let NEWLINE\[NEWLINEH_{\mathrm{top}}(\phi,U)=\lim_{n\to\infty}\frac{\log[U:U_{-n}]}{n},NEWLINE\]NEWLINE where \(U_{-n}=\bigcap_{k=-n}^0 \phi^k(U)\), then NEWLINE\[NEWLINEh_{\mathrm{top}}(\phi)=\sup\{H_{\mathrm{top}}(\phi,U):U\in\mathcal B(G)\}.NEWLINE\]NEWLINE This formula can be used to compare the topological entropy with the scale function.NEWLINENEWLINEThe main result (Theorem 3.1) of the article is NEWLINE\[NEWLINEs_G(\phi)\leq h_{\mathrm{top}}(\phi).NEWLINE\]NEWLINE An example is provided in which the inequality is strict; this is a Bernoulli space of sequences in \(\mathbb Z(p)\), \(p\) prime, indexed by \(\mathbb Z\). Also it is proved under which conditions it has an equality.