Equidistribution from fractal measures
By Michael Hochman, Pablo Shmerkin. 2013
78 citations recorded by the source corpus.
We give a fractal-geometric condition for a measure on $$[0,1]$$[0,1] to be supported on points $$x$$x that are normal in base $$n$$n, i.e. such that $$\{n^kx\}_{k\in \mathbb {N}}$${nkx}k∈N equidistributes modulo 1. This condition is robust under $$C^1$$C1 coordinate changes, and it applies also when $$n$$n is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host’s theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/246005b9d7e46a8fc83bd1bfabd18ac442412569
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, methodology): “orollary of Proposition 3.5 and Lemma 3.6, we obtain the following easy but useful properties concerning typical measures of CP-distributions with positive dimension. Similar results have appeared in [24]. Proposition 3.7. Let P be an ergodic CP-distribution with dimP = h>0. For any ǫ>0, there”