论文标题
使用平面扩展名的分段Möbius间隔图的厄法德证明
Proofs of ergodicity of piecewise Möbius interval maps using planar extensions
论文作者
论文摘要
我们从相关的平面扩展中推导分段Möbius间隔图的动态属性给出了两个结果。 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分段Möbius间隔图的属性。作为概念的证明，我们将这些结果应用于对经过良好研究的Nakada $α$连接的部分的已知结果；我们获得了从无限量的fuchsian群体中得出的间隔图的相似结果。
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.
