Topological mixing, knot points and bounds of topological entropy (Q258248)

The paper under review is motivated by the following conjecture: any continuous nowhere differentiable interval map preserving the Lebesgue measure has infinite topological entropy. NEWLINENEWLINELet \(f\) be a continuous function defined on an interval \(I\). It is well known that the following estimates of the topological entropy holds:NEWLINENEWLINE{\parindent=6mm \begin{itemize}\item[(1)] If \(f\) is is transitive then \(h_{\mathrm{top}}(f)\geq \log(2)/2\) and is attainable. If additionally one of the endpoints is a fixed point then \(f\) is mixing, \(h_{\mathrm{top}} (f) \geq \log 2\) and this value is attainable. \item[(2)] If \(f\) is mixing then \(h_{\mathrm{top}} (f) > \log(2)/2\). \item[(3)] If \(f\) is pure mixing and there is a nonaccessible fixed point, then \(h_{\mathrm{top}} (f) >\log 3\). \item[(4)] If \(f\) is pure mixing and the endpoints of \(I\) form a cycle, then \(h_{\mathrm{top}} (f) >\log(3)/2\).NEWLINENEWLINE\end{itemize}} Additionally, the above lower bounds are the best possible.NEWLINENEWLINEThe authors provide examples in the spirit of Bobok-Soukenka's constructions [\textit{J. Bobok} and \textit{M. Soukenka}, Real Anal. Exch. 36, No. 2, 449--462 (2011; Zbl 1271.37022)] and in other classes of transitive or mixing maps specified in (1)-(4).NEWLINENEWLINEPrecisely, they construct a nowhere monotone map preserving the Lebesgue measure (with dense knot points) which is, respectively, transitive and pure mixing, and has entropy arbitrarily close to the lower bound from Theorem 1.NEWLINENEWLINEWe remind that the knot point of a function \(f \in C(I)\) is any point \(x \in I\) where Dini derivatives satisfy \(D^{+}f(x) = D^{-} f(x) = +\infty\) and \(D_{+}f (x) =D_{-} f (x) = -\infty\).NEWLINENEWLINEThe map \(f\) is said to be transitive if for every pair of non-empty open sets \(U\) and \(V\) there is a non-negative integer \(n\) such that \(f^{(n)}(U)\cap V \neq \emptyset\).NEWLINENEWLINEThe map \(f\) is said to be topologically mixing if for every pair of non-empty open sets \(U\) and \(V\) there is a non-negative integer \(n_0\) such that for every \(n \geq n_0\), \(f^{(n)}(U)\cap V \neq \emptyset\).