Real bounds, ergodicity and negative Schwarzian for multimodal maps
Real bounds, ergodicity and negative Schwarzian for multimodal maps
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DOI：
10.1090/s0894-0347-04-00463-1
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发表时间：
2004-08
影响因子：
3.9
通讯作者：
S. Strien;E. Vargas
中科院分区：
文献类型：
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作者：
S. Strien;E. Vargas
在过去的20年里，动力系统领域中许多最引人注目的结果都是专门处理区间和圆映射（或此类映射的扰动和复杂扩展）的。首先，这是因为在一维情况下，可以获得比一般动力系统更好的失真控制。然而，迄今为止，许多这些壮观的结果只得到了单峰映射。本文的目的是通过获得控制某些首次返回映射的域的几何的 * 真实的界，为研究区间或圆的一般多峰映射提供所有的工具，并提供一个新的（我们相信更简单）证明缺乏游荡区间; * 提供某些组合条件得到满足，大的真实的界意味着某些第一返回映射几乎是线性的;* Koebe distortion控制映射的高迭代的失真，以及某些返回映射的负Schwarzian导数（表明负Schwarzian导数的通常假设是不必要的）; * 控制某些第一返回映射的失真; * 遍历属性，例如遍历分量的数量的严格界限。
Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained so far only for unimodal maps. The aim of this paper is to provide all the tools for studying general multimodal maps of an interval or a circle, by obtaining * real bounds controlling the geometry of domains of certain first return maps, and providing a new (and we believe much simpler) proof of absense of wandering intervals; * provided certain combinatorial conditions are satisfied, large real bounds implying that certain first return maps are almost linear; * Koebe distortion controlling the distortion of high iterates of the map, and negative Schwarzian derivative for certain return maps (showing that the usual assumption of negative Schwarzian derivative is unnecessary); * control of distortion of certain first return maps; * ergodic properties such as sharp bounds for the number of ergodic components.