Dendrites and Chaos

We answer the two questions left open in [Kočan, 2012] i.e. whether there is a relation between [Formula: see text]-chaos and distributional chaos and whether there is a relation between an infinite LY-scrambled set and distributional chaos for dendrite maps. We construct a continuous self-map of a dendrite without any DC3 pairs but containing an uncountable [Formula: see text]-scrambled set. To answer the second question we construct a dendrite [Formula: see text] and a continuous dendrite map without an infinite LY-scrambled set but with DC1 pairs.

Transitive dendrite map with zero entropy

Hoehn and Mouron [Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys.34 (2014), 1897–1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn–Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Entropy estimates for transitive maps on trees. Topology40(3) (2001), 551–569].

DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS

We show that, for monotone graph map f, all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li–Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.