标题： 关于树突上纯混合图的熵 显示英文标题

标题： On entropy of pure mixing maps on dendrites

摘要： 对于每一个$0<\alpha\le\infty$，我们在Gehman分枝线（Gehman dendrite）上构造了一个连续的纯混合映射（拓扑混合，但非精确），其拓扑熵为$\alpha$。 Špitalský此前已证明，在Gehman分枝线上存在拓扑熵任意小的精确映射。 结合这些结果表明，Gehman分枝线上映射的熵并不表现出图映射所报告的那种悖论行为，即精确映射的拓扑熵下确界严格小于纯混合映射的熵下确界。 后一结果用混沌的流行概念表述，意味着图上的映射，较低的熵意味着更强的Devaney混沌。 本文的结论是，对于Gehman分枝线上的映射，较低的熵并不会强制产生更强的混沌。 显示英文摘要

摘要： 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.