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

@article{Brunet2021DimensionsO,
  title={Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers},
  author={Guilhem Brunet},
  journal={Ergodic Theory and Dynamical Systems},
  year={2021}
}
  • 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… 

Figures from this paper

Thermodynamic formalism and large deviation principle of multiplicative Ising models
The aim of this study is tree-fold. First, we investigate the thermodynamics of the Ising models with respect to 2-multiple Hamiltonians. This extends the previous results of [Chazotte and Redig,
Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$
This paper establishs the large deviation principle (LDP) for multiple averages on Nd. We extend the previous work of [Carinci et al., Indag. Math. 2012] to multidimensional lattice Nd for d ≥ 2. The

References

SHOWING 1-10 OF 13 REFERENCES
Hausdorff dimension for fractals invariant under multiplicative integers
Abstract 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
Foundations of Ergodic Theory
Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory suitable for a variety of oneor two-semester courses. The authors’ clear and
Foundations of Ergodic Theory
Preface 1. Recurrence 2. Existence of invariant measures 3. Ergodic theorems 4. Ergodicity 5. Ergodic decomposition 6. Unique ergodicity 7. Correlations 8. Equivalent systems 9. Entropy 10.
Dimensions of some fractals defined via the semigroup generated by 2 and 3
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
Level sets of multiple ergodic averages
We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, the Hausdorff dimensions of the level sets for the limit of
Dimension theory of iterated function systems
Let {Si}  i = 1𝓁 be an iterated function system (IFS) on ℝd with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, 𝓁}. We define the projection entropy function hπ on
Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
TLDR
The objects of ergodic theory -measure spaces with measure-preserving transformation groups- will be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows, and what may be termed the "arithmetic" of these classes of objects is concerned.
Techniques in fractal geometry
Mathematical Background. Review of Fractal Geometry. Some Techniques for Studying Dimension. Cookie-cutters and Bounded Distortion. The Thermodynamic Formalism. The Ergodic Theorem and Fractals. The
Measures of full dimension on affine-invariant sets
Abstract We determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set
...
1
2
...