ABSOLUTELY CONTINUOUS MEASURES FOR CERTAIN MAPS OF AN INTERVAL
ABSOLUTELY CONTINUOUS MEASURES FOR CERTAIN MAPS OF AN INTERVAL
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DOI：
10.1007/bf02698686
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发表时间：
1981-01-01
影响因子：
6.2
通讯作者：
MISIUREWICZ, M
中科院分区：
文献类型：
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作者：
MISIUREWICZ, M
从理论和实验两个角度对区间到自身的映射的动力学性质进行了深入的研究。最重要的理论问题之一(但也与数值结果的可靠性问题密切相关)是确定对于哪些映射存在关于勒贝格测度绝对连续的不变概率度量，它们有多少，以及它们的遍历性质是什么。本文的目的是针对某一类映射回答这些问题。它们本质上是分段单调映射，具有非正的Schwarzian导数，没有下沉和远离临界点的临界点的轨迹(确切的条件可以在第3节：条件(I)-(Vi)中找到)。对于稍相似的一类映射，M.Jakobson[3]证明了绝对连续不变测度的存在性。我们的技术非常不同，它使我们能够获得关于我们的测量的许多信息。在第1-5节的初步结果之后，我们证明了第6节中的主要定理(定理(6.2)和(6.3))。对于我们这类映射，存在有限个(但至少有一个)遍历不变概率测度，它们关于Lebesogue测度是绝对连续的。它们的密度在开密集上是连续的。在映射的第n次迭代下(对于某个k)，关于勒贝格测度绝对连续的每个有限测度的像都强收敛(作为n-gt；co)到这些测度的线性组合。在第7节中，我们证明了对于大多数被广泛考虑的单参数映射族(如^1-^40^(1-^))，我们的条件((I)-(Vi))对于幂连续统的参数集是满足的。这组参数的衡量标准是零还是正，这个问题仍然悬而未决。然而，有一些证据表明，
Dynamical properties of mappings of an interval into itself are intensely studied from both< c theoretical" and c< experimental5?(numerical experiments) points of view. One of the most important theoretical problems (but also closely related to the problem of reliability of numerical results) is to establish for which mappings there exist invariant probabilistic measures, absolutely continuous with respect to the Lebesgue measure, how many of them, and what are their ergodic properties. The aim of this paper is to answer these questions for a certain class of mappings. They are essentially the piecewise monotone mappings with non-positive Schwarzian derivative, no sinks and trajectories of critical points staying far from critical points (the exact conditions can be found in section 3: conditions (i)-(vi)). For a slightly similar class of mappings M. Jakobson [3] proved the existence of an absolutely continuous invariant measure. Our technique is quite different and it enables us to obtain much information about our measuies. After the preliminary results of Sections 1-5, we prove the main theorems (theorems (6.2) and (6.3)) in Section 6. For a mapping from our class, there exist a finite number (but at least one) of ergodic invariant probabilistic measures, absolutely continuous with respect to the Lebesgue measure. Their densities are continuous on an open dense set. Images of every finite measure, absolutely continuous with respect to the Lebesgue measure, under nk-th iterations of the mapping (for a certain k), converge strongly (as n-> co) to a linear combination of those measures. The mapping with every one of those measures is a skew product of a permutation of a finite set (in the base) and an exact transformation.In Section 7 we show that for most widely considered one-parameter families of mappings (like^ 1-^ 40^(1—^)), our conditions ((i)-(vi)) are satisfied for a set of parameters of power the continuum. The question, whether the measure of this set of parameters is zero or positive, remains open. However, there is some evidence that