arXiv 1709.010452017-09-04math.DS

混沌耗散边界的无穷多稳定模

Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos

Peter Hazard, Marco Martens, Charles Tresser

备注：: 36 pages, 3 figures. Accepted for publication in Transactions of the American Mathematical Society

摘要：

在具有零拓扑熵的面积收缩Hénon-like映射族中，我们证明了存在具有无穷多个稳定模的映射.因此，在一个具有多个参数的映射族中，不可能找到所有可能的非混沌收缩面积Hénon-like映射的拓扑型.一个类似的结果，但在家庭的混乱地图，成为民间传说的一部分后不久，埃农使用这样的地图，以产生什么很快被证实是第一个非双曲奇怪吸引子在$\mathbb{R}^2$。我们的证明使用了最近关于可无限重正化的面积收缩的Hénon-like映射的结果;它表明，对于其周期轨道的周期集是有限的（特别是等于$\{1,\, 2,\dots,\,2^{n-1}\}$或这个$n$元组的初始段）的面积收缩的Hénon-like映射，表示所有可能的拓扑型所需的参数的数量随着周期的数量而增加。相比之下，在2-圆盘与$k\geq 1$的$C^k$ -嵌入中，零熵集内部的非混沌但非面积收缩映射的最大模数是无穷大的。

英文摘要：

In the family of area-contracting Hénon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ or an initial segment of this $n$-tuple) increases with the number of periods. In comparison, among $C^k$-embeddings of the 2-disk with $k\geq 1$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.

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