Syndetically proximal pairs
Consider a dynamical system formed by the complex \((X,f)\) where \(X\) is a compact metric space and \(f: X\to X^-\) a continuous map. The aim of this paper is to analyze the concept of syndetically proximal as an extension of the classical notion. Sufficient conditions for having syndetically pairs are given. It is proven that any interval map \(f\) with positive topological entropy has a syndetically scrambled Cantor set and an uncountable syndetically scrambled set invariant under some power of \(f\). Moreover, an improvement is obtained of the classical result: ``If \(f\) is an interval map with positive topological entropy and \(m\geq 2\) then there is \(n\in\mathbb{N}\) such that the one-sided full shift on \(m\) symbols is semi-conjugate to a subsystem of \(f^{2^{n}}\)'' by stating conjugacy instead of semi-conjugacy.
scientific article; zbMATH DE number 5880879
scientific article