On entropy of pure mixing maps on dendrites

TL;DR

The paper investigates the range of topological entropy values for maps on the Gehman dendrite within the transitive, mixing, and exact hierarchy, addressing whether an entropy paradox exists. It constructs, for each $h_0\in(0,\infty]$, a pure mixing map $F$ with $h(F)=h_0$ by a floor-wise assembly of exact Markov tree maps on an engineered Gehman dendrite, and then refines to a globally defined, $\sigma$-linear expanding map under a geodesic metric with $\dim_H(\mathcal{G})=1$. The main contribution is showing that pure mixing maps can realize every entropy value in $(0,\infty]$, and together with Špitalský’s result on arbitrarily small positive entropy for exact maps, that the infima of entropies over transitive, pure mixing, and exact maps on the Gehman dendrite all coincide at $0$, i.e., no entropy paradox occurs on this dendrite. This work highlights entropy-flexibility on a one-dimensional continuum and delineates how chaos notions interact differently than on graphs.

Abstract

For every $0<α\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $α$. It has been previously shown by Špitalský that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.

Figures (3)

• Figure 1: The tree $T^{(3)}$ with the standard labelling of its vertices.
• Figure 2: The action of $\omega\mapsto \omega\oplus 1$ operation on binary words of length $3$.
• Figure 3: The action of $G$ on the vertices of $T^{(3)}$.

Theorems & Definitions (20)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4: Misiurewicz_hausdorff
• Definition 2.5
• Theorem 2.6
• Corollary 2.7
• Theorem 3.1
• proof
• Claim 1