Hausdorff dimension for fractals invariant under the multiplicative integers

Abstract

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k{x}_{2k}=0$ for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

22 pages, 2 figures; revision after the referee report: corrected minor typos, Example 3.4 expanded, added concluding remarks and references in section 6; accepted for publication in Ergodic Theory and Dynamical Systems

References in corpus (1)

• Level sets of multiple ergodic averages

Cited by in corpus (1)

• Multifractal analysis of some multiple ergodic averages for the systems with non-constant Lyapunov exponents