Invariance of multifractal spectrum of uniform self-affine measures and its applications

Abstract

We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let ${\mu }_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling property of ${\mu }_E$ are both invariant under a bi-Lipschitz map. Moreover, we show that if ${\mu }_E$ and ${\mu }_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a certain measure preserving property.

References in corpus (1)

• Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Cited by in corpus (1)

• Relations between topological and metrical properties of self-affine Sierpi$\acute{\text{n}}$ski sponges