Dynamics of homeomorphisms of regular curves
Volume 162 / 2020
Abstract
Let $f:X\to X$ be a homeomorphism of a regular curve $X$. We prove that the space of minimal sets of $f$ is closed in the hyperspace $2^X$ of closed subsets of $X$ endowed with the Hausdorff metric. As a consequence, we establish the equivalence between pointwise periodicity of $f$ and the Hausdorffness of the orbit space $X/f$. Moreover, we prove that the nonwandering set $\Omega (f)$ is equal to the set of recurrent points of $f$ and we study the continuity of the map $\omega _f:X\to 2^X$, $x\mapsto \omega _f(x)$. We show for instance the equivalence between the continuity of $\omega _f$ and the equality between the $\omega $-limit set and the $\alpha $-limit set of every point in $X$. Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.
