Abstract
We study the scaling scenery and limit geometry of invariant measures for the nonconformal toral endomorphism (x, y) → (mx mod 1, ny mod) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CPchain in the sense of Furstenberg. Invoking the machinery of CPchains 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 CPchains and hence verify it for various classes of fractals such as selfaffine carpets of Bedford–McMullen, Lalley–Gatzouras and Barański class and all planar selfsimilar sets.
Original language  English 

Pages (fromto)  564602 
Number of pages  39 
Journal  Advances in Mathematics 
Volume  268 
Early online date  22 Oct 2014 
DOIs  
Publication status  Published  2 Jan 2015 
Keywords
 CPchain
 Symbolic dynamics
 Bernoulli measure
 Hausdorff dimension
 Selfaffine carpet
 Projections
 The distance set conjecture
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Jonathan Fraser
 School of Mathematics and Statistics  Director of Research
 Pure Mathematics  Professor
Person: Academic