Every nondegenerate Peano continuum admits a topologically mixing but not exact self-map with dense periodic points.
On entropy of pure mixing maps on dendrites
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abstract
For every $0<\alpha\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $\alpha$. It has been previously shown by \v{S}pitalsk\'y 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.
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UNVERDICTED 2representative citing papers
Continuous transitive non-minimal maps on compact metric spaces with a free interval are relatively mixing, non-invertible, have positive topological entropy, and dense periodic points.
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Every nondegenerate Peano continuum admits a pure mixing selfmap
Every nondegenerate Peano continuum admits a topologically mixing but not exact self-map with dense periodic points.
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On mixing and dense periodicity on spaces with a free arc
Continuous transitive non-minimal maps on compact metric spaces with a free interval are relatively mixing, non-invertible, have positive topological entropy, and dense periodic points.