Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6425623 | Advances in Mathematics | 2015 | 39 Pages |
Abstract
We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x,y)â¦(mxmod1,nymod1) 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 BaraÅski class and all planar self-similar sets.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Andrew Ferguson, Jonathan M. Fraser, Tuomas Sahlsten,