Indexed on: 14 Sep '18Published on: 14 Sep '18Published in: arXiv - Mathematics - Dynamical Systems
In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. In particular, we prove that every invariant measure of quasi-graph map with zero topological entropy has discrete spectrum. We also obtain an analog of Llibre-Misiurewicz's result relating positive topological entropy with existence of topological horseshoes. By these results, Sarnak's M\"obius Disjointness Conjecture restricted to the class of quasi-graph maps with zero topological entropy is reduced to already known cases. We prove however, that answering the conjecture for all maps on dendrites with zero topological entropy is equivalent to solving it for all dynamical systems with zero topological entropy.