%0 Unpublished work %T DIMENSIONS OF “SELF-AFFINE SPONGES” INVARIANT UNDER THE ACTION OF MULTIPLICATIVE INTEGERS %+ Laboratoire Analyse, Géométrie et Applications (LAGA) %A Brunet, Guilhem %8 2021-11-09 %D 2021 %Z 2010.03230 %K Hausdorff dimension %K Box dimension %K Symbolic dynamics %K Self-affine carpets %K Self-affine sponges %Z Mathematics [math]/Dynamical Systems [math.DS]Preprints, Working Papers, ... %X 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. %G English %2 https://sorbonne-paris-nord.hal.science/hal-02958411v3/document %2 https://sorbonne-paris-nord.hal.science/hal-02958411v3/file/ETDS_V5.pdf %L hal-02958411 %U https://sorbonne-paris-nord.hal.science/hal-02958411 %~ UNIV-PARIS13 %~ UNIV-PARIS8 %~ CNRS %~ LAGA %~ INSMI %~ TDS-MACS %~ UNIV-PARIS-LUMIERES %~ SORBONNE-PARIS-NORD %~ UNIV-PARIS8-OA