On the topological entropy of continuous and almost continuous functions (Q719733)

The paper under review can be divided into two parts. In the first part the authors introduce new notions of an entropy point and a strong entropy point for a function \(f:X\to X\), where \(X\) is a compact metric space. (Note that those definitions are different from other notions of entropy point which occur in the literature, because they are connected with the entropy of \(f\)-bundles. In particular, each strong entropy point is a fixed point of \(f\).) In the main theorem of this part the authors show that if \(f,g:X\to X\) are conjugate then \(f\) has an entropy point (a strong entropy point) iff \(g\) has the same property. In the second part the authors consider approximations of a function \(f:X\to X\) (here \(X\) is a compact manifold) by functions having strong entropy points. Two kinds of approximation are considered: the \(\Gamma\)-approximation and the uniform approximation. Recall that a function \(g:X\to X\) is \(\Gamma\)-approximated by functions belonging to the class \(\mathcal{F}\) if for each open set \(U\subset X\times X\) containing the graph of \(g\) there exists \(f\in\mathcal{F}\) with the graph contained in \(U\). (In particular, \(g:X\to X\) is \(\Gamma\)-approximated by continuous functions iff it is almost continuous in the sense of Stallings.) The main results of this part: every almost continuous function can be \(\Gamma\)-approximated by (almost) continuous functions having strong entropy points; and every continuous function can be uniformly approximated by continuous functions having strong entropy points.