利用平面扩展证明片断莫比乌斯区间映射的遍历性

IF 0.8 4区 数学 Q2 MATHEMATICS Expositiones Mathematicae Pub Date : 2024-05-15 DOI:10.1016/j.exmath.2024.125575
Kariane Calta , Cor Kraaikamp , Thomas A. Schmidt
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引用次数: 0

摘要

我们给出了从片断莫比乌斯区间映射的相关平面扩展推导其动力学性质的两个结果。首先,平面扩展的温和有限性条件以及用于放宽传统马尔可夫条件的新特性 "有界非全范围",都会导致最终扩张性和等同于勒贝格度量的遍历不变概率度量的存在。其次,由 Kraaikamp 和合著者引入的对适当邻近平面系统的 "绗缝 "操作可用于证明片断莫比乌斯区间图的几个关键动力学性质。作为概念证明,我们应用这些结果恢复了对中田 α 连续分数的已知研究结果;我们还获得了从不可通约的福氏群无穷族衍生出的区间映射的类似结果。
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Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada α-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
41
审稿时长
40 days
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