Scaling scenery of (x m, x n) invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal 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 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.