Entropy of induced maps of regular curves homeomorphisms

Abstract: Let $f:X\to X$ be a self homeomorphism of a continuum $X$, we show that the topological entropy of the induced system $(2^X,2^f)$ is infinite provided that $X\setminus \Omega(f)$ is not empty. If furthermore $X$ is a regular curve then it is shown that $(2^X,2^f)$ has infinite topological entropy if and only if $X\setminus \Omega(f)$ is not empty. Moreover we prove for the induced system $(C(X),C(f))$ the equivalence between the following properties: (i) zero topological entropy; (ii) there is no Li-Yorke pair and (iii) for any periodic subcontinnum $A$ of $X$ and any connected component $C$ of $X\setminus \Omega(f)$, $C\subset A$ if $A\cap C\neq \emptyset$. In particular, the topological entropy of either $(2^X,2^f)$ or $(C(X),C(f))$ has only two possible values $0$ or $\infty$. At the end, we give an example of a pointwise periodic rational curve homeomorphism $F:Y\to Y$ with infinite topological entropy induced map $C(F)$.