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The Dimension of Integral Self-Affine Sets via Fractal Perturbations: The Box and the Hausdorff Dimensions, Ergodic Measures

Ibrahim Kirat

Abstract

An integral self-affine set $F=F(T,A)\subseteq \mathbb{R}^n$ is a self-affine set which is generated by an $n\times n$ integer expanding matrix $T$ (not necessarily a similitude) and a finite set $A\subset \mathbb{Z}^n$ of integer vectors so that $F=T^{-1}(F+A)$. The dimension problem of $F$ has not yet been settled fully. For that, we introduce a fractal perturbation method with respect to $T,A$ and get the dimension as the limit of the dimensions of a sequence of better-behaved perturbed fractals, for which a dimension formula already exists. An unexpected feature of this technique is that the overlap structures of $F$ and its perturbations are eventually the same (i.e. the neighbor graphs are isomorphic), which is unlike some known perturbations. Our method has been developed especially for the problematic case of irreducible characteristic polynomial of $T$. Also, we do not impose any separation condition on $F$ (like the open set condition) or any further restriction (such as size, etc.) on $T$ or $A$. As a by-product of the perturbation method, we prove the existence of the box dimension of $F$ too. Further, we consider $F$ as a $T$-invariant subset of the n-torus (i.e, we consider $F \ \rm{mod \ 1}$), and we rather use the perturbation method to show that there is an ergodic $T$-invariant Borel probability measure on $F \ \rm{mod \ 1}$ of full dimension. In contrast to some known results, this is not an almost-sure result.

The Dimension of Integral Self-Affine Sets via Fractal Perturbations: The Box and the Hausdorff Dimensions, Ergodic Measures

Abstract

An integral self-affine set is a self-affine set which is generated by an integer expanding matrix (not necessarily a similitude) and a finite set of integer vectors so that . The dimension problem of has not yet been settled fully. For that, we introduce a fractal perturbation method with respect to and get the dimension as the limit of the dimensions of a sequence of better-behaved perturbed fractals, for which a dimension formula already exists. An unexpected feature of this technique is that the overlap structures of and its perturbations are eventually the same (i.e. the neighbor graphs are isomorphic), which is unlike some known perturbations. Our method has been developed especially for the problematic case of irreducible characteristic polynomial of . Also, we do not impose any separation condition on (like the open set condition) or any further restriction (such as size, etc.) on or . As a by-product of the perturbation method, we prove the existence of the box dimension of too. Further, we consider as a -invariant subset of the n-torus (i.e, we consider ), and we rather use the perturbation method to show that there is an ergodic -invariant Borel probability measure on of full dimension. In contrast to some known results, this is not an almost-sure result.
Paper Structure (15 sections, 42 theorems, 353 equations)

This paper contains 15 sections, 42 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Dynamically Perturbed (Deflected) Fractals
  3. Reduction to Triangular Matrices
  4. Perturbed (Deflected) Fractals
  5. Basic Results
  6. Neighbor Structure of Dynamical Perturbations
  7. An Application: The Box Dimension
  8. The Hausdorff Dimension
  9. Tile Measures: A set-up needed for the Hausdorff dimension
  10. Proof of Theorem \ref{['Hausdorff']}: First Part
  11. Lower Bounds for Dimension
  12. Another Application: Measures of Full Dimension
  13. Appendix : Dimension Formulas for Perturbed Fractals $F_k$
  14. Deconstruction of $F$
  15. The Box Dimension of $F_k$

Key Result

Theorem 1.1

Let $F\subset \mathbb{R}^n$ be an integral self-affine set and $\delta_{k}=dim_H F_{k}$. Then Further, there is a positive integer $k_0$ such that $\delta_{k} \leq dim_H F$ for every $k\geq k_0$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 47 more