Lipschitz classification of Bedford-McMullen carpets (I): Invariance of Multifractal spectrum and arithmetic doubling property

Hui Rao, Ya-min Yang, Yuan Zhang

Published 2020-05-15Version 1

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 of $\mu_E$ is a bi-Lipschitz invariant, and the doubling property of $\mu_E$ is also invariant under a bi-Lipschtz map. We show that if $E$ and $F$ are totally disconnected and that $\mu_E$ and $\mu_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a measure preserving property. Thanks to the above results, we give a complete classification of Bedford-McMullen carpets which are regular(that is, its Hausdorff dimension and box dimension coincides,) or satisfy a separation condition due to [J. F. King, The Singularity spectrum for general sierpinski carpets, \textit{Adv. Math.} \textbf{116} (1995), 1-11].