Hausdorff dimension for fractals invariant under the multiplicative integers

Details Hausdorff dimension for fractals invariant under the multiplicative integers Hausdorff dimension for fractals invariant under the multiplicative integers

2011-02-25

arxiv.org/abs/1102.5136v2

Mathematics

Dynamical Systems

22 pages, 2 figures; revision after the referee report: corrected minor typos, Example 3.4 expanded, added concluding remarks

Scientific paper

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.

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