论文标题
邀请粗糙动态的邀请：拉链地图
An invitation to rough dynamics: zipper maps
论文作者
论文摘要
在动态系统的领域中，遇到不规则函数并不罕见，通常是h {Ö} lder，而不是Lipschitz（例如WeierStrass函数）。我们的目标是刮擦以下问题的表面：如果我们考虑不规则地图并迭代它们，会发生什么？我们介绍了“拉链地图”的家族，在上述意义上是不规则的，并研究了它们的一些动力学特性。对于大量参数，相应的拉链地图允许所有订单的马蹄铁；直接的结果，$ k \ ell $点上的每个订单都可以由地图的$ k $ orbits实现消失的绝对度量平均维度，证明了一个猜想的小情况，即绝对度量平均维度与拓扑平均维度一致。
In the field of dynamical systems, it is not rare to meet irregular functions, which are typically H{ö}lder but not Lipschitz (e.g. the Weierstrass functions). Our goal is to scratch the surface of the following question: what happens if we consider irregular maps and iterate them? We introduce the family of "zipper maps", which are irregular in the above sense, and study some of their dynamical properties. For a large set of parameters, the corresponding zipper map admits horseshoe of all orders; as an immediate consequence, every order on $k\ell$ points can be realized by $k$ orbits of length $\ell$ of the map.These maps have infinite topological entropy, and we refine this statement by showing that they have positive metric mean dimension with respect to the Euclidean metric, as well as by introducing other notions of higher complexity.Finally, we prove that every interval map (thus including zipper maps) have vanishing absolute metric mean dimension, proving a small case of the conjecture that the absolute metric mean dimension coincides with the topological mean dimension.
