Hausdorff Dimension for Fractals Invariant under the Multiplicative Integers
نویسندگان
چکیده
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 (xk) such that xkx2k = 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.
منابع مشابه
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We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σm = {0, ..., m−1} N that are invariant under multiplication by integers. The results apply to the sets {x ∈ Σm : ∀ k, xkx2k · · ·xnk = 0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.
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