Scaling scenery of $(\times m,\times n)$ invariant measures
Scaling scenery of $(\times m,\times n)$ invariant measures
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$( imes m, imes n)$ 不变度量的缩放场景
DOI：
10.1016/j.aim.2014.09.019
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Tuomas Sahlsten
中科院分区：
文献类型：
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作者：
Andrew Ferguson;J. Fraser;Tuomas Sahlsten
本文研究了自然Markov划分的Bernoulli测度的非共形toral自同态（x，y）<$（mxmod 1，nymod 1）的不变测度的标度风景和极限几何.我们表明，统计的标度可以描述的遍历CP-链的意义上的Furstenberg。对CP-链的构造进行简化，得到了伯努利测度的投影定理，它部分地推广了Hochman-Shmerkin和Ferguson-Jordan-Shmerkin的早期结果。利用CP-链给出了法尔科纳距离集猜想维数部分的遍历性理论判据，并对Bedford-McMullen类、Lalley-Gatzouras类和Barauburski类的自仿射地毯以及所有平面自相似集等各种分形类进行了验证.
We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x, y)↦(m x mod 1, n y mod 1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman–Shmerkin and Ferguson–Jordan–Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford–McMullen, Lalley–Gatzouras and Barański class and all planar self-similar sets.