Hausdorff dimension of the limit sets of some planar geometric constructions

Barański, Krzysztof

doi:10.1016/j.aim.2006.06.005

Cited by 94 publications

(161 citation statements)

References 12 publications

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“…Self-affine attractors are intensively studied, and many results are known about some particular families. For example the Hausdorff dimension of BedfordMcMullen carpets admits an exact simple formula [24,74], and similar results about the fractal dimension or the Lebesgue measure of some other classes exist [18,35,47,50,64]. Moreover, there is an "almost sure" formula for the packing and Hausdorff dimension in the self-similar case [44].…”

Section: Undecidability Of Gifs Topological Propertiesmentioning

confidence: 99%

Decidability Problems for Self-induced Systems Generated by a Substitution

Jolivet

Siegel

2015

Lecture Notes in Computer Science

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Abstract. In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.The following survey is mainly inspired by three papers from the authors and their collaborators [32,60,84].

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Section: Undecidability Of Gifs Topological Propertiesmentioning

confidence: 99%

Decidability Problems for Self-induced Systems Generated by a Substitution

Jolivet

Siegel

2015

Lecture Notes in Computer Science

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“…[23,Corollary 9.1.7]), but it is far more difficult to compute the Hausdorff dimension of even relatively simple non-conformal fractals, such as the limit sets/measures of affine iterated function systems (IFSs) satisfying the open set condition. To make progress one generally has to assume either some randomness in the contractions defining the IFS, as in [11,16], or some special relations between these contractions, as in [1,7]. An exception to this is a recent theorem of Bárány and Käenmäki [2], who showed that every self-affine measure on the plane is exact dimensional.…”

Section: T Das Et Almentioning

confidence: 99%

“…The class of self-affine sponges is the generalization to higher dimensions of the class of self-affine carpets, which consists of subsets of R 2 defined according to a certain recursive construction where each rectangle in the construction is replaced by the union of several rectangles contained in that rectangle (see Definition 1.1 below). The Hausdorff dimension of certain self-affine carpets was computed independently by Bedford [3] and McMullen [22], and their results were extended by several authors [1,7,18,21].…”

Section: T Das Et Almentioning

confidence: 99%

Badly approximable points on self-affine sponges and the lower Assouad dimension

Das¹,

Fishman²,

Simmons³

et al. 2017

Ergod. Th. Dynam. Sys.

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Abstract. We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.Fix d ∈ N. Dirichlet's theorem in Diophantine approximation states that, for all x ∈ R d , there exist infinitely many rational points p/q ∈ Q d such thatA point x ∈ R d is said to be badly approximable if this inequality cannot be improved by more than a constant, i.e. if there exists a constant c > 0 such that, for any rational

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“…Dimensions of self-affine sets are more awkward to analyse, [2,5,6,11,14]. Very little has been done to extend this theory to the random setting: working in a submultiplicative rather than a multiplicative setting presents many challenges.…”

Section: Introductionmentioning

confidence: 99%