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DIMENSIONS OF “SELF-AFFINE SPONGES” INVARIANT UNDER THE ACTION OF MULTIPLICATIVE INTEGERS

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Résumé

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots,m_1-1\} \times \{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres and Solomyak and using a fixed integer $q \geq 2$. We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpinski carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \left\lfloor \log_q \left( \frac{\log(m_1)}{\log(m_2)} \right) \right\rfloor$. We then generalize our results to the same subsets defined in dimension $d \geq 2$. There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.
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Dates et versions

hal-02958411 , version 1 (06-10-2020)
hal-02958411 , version 2 (02-10-2021)
hal-02958411 , version 3 (09-11-2021)

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Guilhem Brunet. DIMENSIONS OF “SELF-AFFINE SPONGES” INVARIANT UNDER THE ACTION OF MULTIPLICATIVE INTEGERS. 2021. ⟨hal-02958411v3⟩
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