On pairwise sensitive homeomorphisms

TL;DR

This paper studies pairwise sensitive homeomorphisms, showing their measure-zero semi-orbit points, non-existence on compact intervals, and characterization as Denjoy homeomorphisms on circles, with applications to expansive systems.

Contribution

It provides new properties of pairwise sensitive homeomorphisms, including measure-theoretic and topological characterizations, and offers alternative proofs for known results in expansive dynamics.

Findings

Points with converging semi-orbits have measure zero

Such homeomorphisms do not exist on compact intervals

On circles, they are exactly the Denjoy homeomorphisms

Abstract

We obtain properties of the pairwise sensitive homeomorphisms defined in \cite{cj}. For instance, we prove that their sets of points with converging semi-orbits have measure zero, that such homeomorphisms do not exist in a compact interval and, in the circle, they are the Denjoy ones. Applications including alternative proofs of well-known facts in expansive systems are given.