The Fibonacci unimodal map
The Fibonacci unimodal map
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DOI：
10.1090/s0894-0347-1993-1182670-0
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发表时间：
1991-08
影响因子：
3.9
通讯作者：
M. Lyubich;J. Milnor
中科院分区：
文献类型：
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作者：
M. Lyubich;J. Milnor
临界轨道的斐波那契递推出现在 Branner 和 Hubbard 关于复三次多项式的工作中 [BH, §12] 以及 Yoccoz 关于二次多项式的工作 [Y1, Y2] 中，作为递推的“最差”模式。另一方面，Hofbauer 和 Keller [HK] 建议将实二次斐波那契图作为具有“狂野”吸引子的图的可能候选者（即，集合 A，它是勒贝格几乎每个轨道的 w 极限集，但严格小于通用轨道的 w 极限集）。 [HK] 中临界轨道的 w 极限集拥有所有已知的野吸引子拓扑特性（比较 [BL2]）。事实上，我们将在下面看到二次斐波那契图没有野吸引子；然而，具有退化临界点的地图的相应问题仍然悬而未决。实际上，斐波那契图的第一个迹象出现在 Tsuda [T] 的数值著作中，与 Belousova-Zhabotinskii 反应相关，以及 Shibayama [Sh] 的数值著作（更准确地说，他们研究了创建斐波那契图的“斐波那契分叉”序列）。本文将研究真实斐波那契图的拓扑、几何和测度理论特性。我们的目标是弄清楚这种类型的复发是否确实给出了任何病理学例子，并将其与沙利文[S]研究的无限可重整化复发模式进行比较。事实证明，这种情况是完全可以理解的，并且是非常有规律的。特别是，任何斐波那契图（具有负施瓦茨和非简并临界点）都具有绝对连续的不变测度（因此，我们处理“常规”类型的混沌动力学）。事实证明，临界轨道闭合的几何特性与费根鲍姆映射的几何特性有很大不同：它的豪斯多夫维数等于零，并且它的几何形状不是刚性的，而是取决于一个参数。布兰纳和哈伯德引入了画面的概念来描述临界轨道的重现。他们的“斐波那契图表”是一个基本的例子，它对应于一种特别紧密且有规律的重现模式。如果一个复二次映射 z 1-+ z2 + c 实现了这个斐波那契表格，那么
The Fibonacci recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the "worst" pattern of recurrence. On the other hand, a real quadratic Fibonacci map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a "wild" attractor (that is, a set A which is the w-limit set for Lebesgue almost every orbit but is strictly smaller than the w-limit set for a generic orbit). The w-limit set of the critical orbit in [HK] possesses all known topological properties of wild attractors (compare [BL2]). In fact, we will see below that the quadratic Fibonacci map does not have a wild attractor; however, the corresponding question for a map with a degenerate critical point remains open. Actually, the first indication of the Fibonacci map appeared in the numerical work of Tsuda [T], related to the Belousova-Zhabotinskii reaction, and also in numerical work of Shibayama [Sh] (more precisely, they studied the sequence of "Fibonacci bifurcations" creating the Fibonacci map). This paper will study topological, geometrical, and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan [S]. It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure (so, we deal with a "regular" type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter. Branner and Hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits. Their "Fibonacci tableau" is a basic example, which corresponds to one particularly close and regular pattern of recurrence. If a complex quadratic map z 1-+ z2 + c realizes this Fibonacci tableau, then the