Space-filling curves and geodesic laminations. II: Symmetries

Sirvent, Vı́ctor F.

doi:10.1007/s00605-012-0406-9

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…But in this method, the self-similar sets are not given a priori. Other attempts of constructing substitution rules for special self-similar sets can be found in Remes [39] and Sirvent [42][43][44].…”

Section: A Brief History Of Space-filling Curvesmentioning

confidence: 99%

Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

2019

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It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper.Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the positive Euler-tour class. Finally, we propose a general Euler-tour method, using which we show that if a self-similar set satisfies the open set condition and possesses a skeleton, then space-filling curves can be constructed. Especially, all connected selfsimilar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.MSC 2000: 28A80, 54C05.

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Section: A Brief History Of Space-filling Curvesmentioning

confidence: 99%

Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

2019

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“…The topology of Julia sets, and of the Mandelbrot set, can be described by the knowledge which rays land together (the Carathéodory loop), at least under the assumption of local connectivity. Mating constructions [R,Sh,Ta2] and certain constructions of space-filling curves [Si1,Si2] also rely on the biaccessibility of external angles (and rays). In addition, at least for postcritically finite parameters or more generally, parameters with a compact Hubbard tree [Ti], the Hausdorff dimension of the set of biaccessible dynamical angles is, up to a factor log 2, equal to the core entropy, i.e., the entropy of the map restricted to the Hubbard tree.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of biaccessible angles for quadratic polynomials

Bruin¹,

Schleicher²

2017

Fund. Math.

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A point c in the Mandelbrot set is called biaccessible if two parameter rays land at c. Similarly, a point x in the Julia set of a polynomial z → z 2 + c is called biaccessible if two dynamic rays land at x. In both cases, we say that the external angles of these two rays are biaccessible as well.In this paper we will use a purely combinatorial characterization of biaccessible (both dynamic and parameter) angles, and use it to give detailed estimates of the Hausdorff dimension of the set of biaccessible angles.

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Self-similar sets satisfying the common point property

Sirvent

2014

Chaos, Solitons & Fractals

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Space-filling curves and geodesic laminations. II: Symmetries

Cited by 3 publications

References 12 publications

Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

Hausdorff dimension of biaccessible angles for quadratic polynomials

Self-similar sets satisfying the common point property

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