Ibrahim Kirat | Searcharxiv

arXiv2026

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

Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original 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.

math.DS↗

arXiv2019

On the Convex Hulls of Self-Affine Fractals

Suppose that the set ${\mathcal{T}}= \{T_1, T_2,...,T_q \} $ of real $n\times n$ matrices has joint spectral radius less than $1$. Then for any digit set $ D= \{d_1, \cdots, d_q\} \subset {\Bbb R}^n$, there exists a unique nonempty compact set $F=F({\mathcal{T}},D)$ satisfying $ F = \bigcup _{j =1}^q T_j(F + d_j)$, which is called a self-affine fractal. We consider an existing criterion for the convex hull of $F$ to be a polytope, which is due to Kirat and Kocyigit. In this note, we strengthen our criterion for the case $T_1=T_2=\cdots =T_q $. More specifically, we give an upper bound for the number of steps needed for deciding whether the convex hull of $F$ is a polytope or not. This improves our earlier result on the topic.

math.DS↗

arXiv2016

On a Topological Problem of Strange Attractors

Somehow, the revised version of our paper \cite{KY} does not appear on journals' home page. Here we present the revised version altered to reflect the corrections and/or additions to that paper. In this note, we consider self-affine attractors that are generated by an integer expanding $n\times n$ matrix (i.e., all of its eigenvalues have moduli $>1$) and a finite set of vectors in ${\Bbb{Z}}^n$. We concentrate on the problem of connectedness for $n\leq 2$. Although, there has been intensive study on the topic recently, this problem is not settled even in the one-dimensional case. We focus on some basic attractors, which have not been studied fully, and characterize connectedness.

math.DS↗