On the relation between topological entropy and entropy dimension (Q736232)

Let \(X\) be a compact metric space of finite entropy dimension, \(f:X\to X\) be a Lipschitz map, and \(\mathbb{D}\) be the set of metrics on \(X\) that generate the topology on \(X\). It is known that the topological entropy of \(f\) has the following upper bound: \[ h_{\mathrm{top}}(f)\leq\inf_{d\in\mathbb{D}}\dim_E(X,d)L_d(f), \] where \(\dim_E(X,d)\) is the entropy dimension of \(X\) relative to \(d\), \(L_d(f)\) is the Lipschitz constant of \(f\) relative to \(d\) (see [\textit{A. Katok} and \textit{B. Hasselblatt}, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, Cambridge (1997; Zbl 0878.58019)]). The paper under review is devoted to the study of necessary and sufficient conditions for the above inequality to be an equality. The author of the paper asserts that this is a conjecture of E. Ghys. For \(d\in\mathbb{D}\), let \(\Sigma_d(N,\varepsilon)\) denote the minimal cardinality of an \((N,\varepsilon)\)-net for \(d\). The convergence in the definition of \(h_{\mathrm{top}}(f)\) is said to be exponential if for any \(\delta>0\), there exist \(d_\delta\in\mathbb{D}\) and \(C,\varepsilon_\delta>0\) such that \[ \frac{\ln\Sigma_{d_\delta}(N,\varepsilon)}{N}\leq h_{\mathrm{top}}(f)+\delta \] whenever \(0<\varepsilon<\varepsilon_\delta\) and \(N>C|\ln\varepsilon|\). The author proves that Ghys' conjecture holds if and only if the convergence in the definition of \(h_{\mathrm{top}}(f)\) is exponential. As an application, the author shows that if \(f\) is an intrinsically hyperbolic Lipschitz homeomorphism in the sense of [\textit{Y. Ilyashenko} and \textit{W. Li}, Nonlocal bifurcations. Mathematical Surveys and Monographs 66, American Mathematical Society, Providence, RI (1999; Zbl 1120.37308)], then Ghys' conjecture holds.