Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

  title={Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers},
  author={Guilhem Brunet},
  journal={Ergodic Theory and Dynamical Systems},
  • Guilhem Brunet
  • Published 7 October 2020
  • Mathematics
  • Ergodic Theory and Dynamical Systems
Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ 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… 

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