arXiv subjects
Publications and source records attributed to Ghassen Askri.
We prove that the Möbius disjointness conjecture holds for graph maps and for all monotone local dendrite maps. We further show that this also hold for continuous map on certain class of dendrites. Moreover, we see that there is a transitive dendrite map with zero entropy for which Möbius disjointness holds.
Let $X$ be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $ω$-limit set of $f$ then $L\cap P(f)\subset E(X)^{\prime}$, where $E(X)^{\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\cap P(f)=\emptyset$. We also show that if $E(X)^{\prime}$ is finite then any uncountable $ω$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yorke pair $(x,y)$ with $ω\_f(x)$ or $ω\_f(y)$ is uncountable then $f$ is Li-Yorke chaotic.