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Parabolic Cantor sets

Mariusz Urbański (1996)

Fundamenta Mathematicae

The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the...

Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes

Krzysztof Gdawiec, Diana Domańska (2011)

International Journal of Applied Mathematics and Computer Science

One of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from...

Poincaré inequality and Hajłasz-Sobolev spaces on nested fractals

Katarzyna Pietruska-Pałuba, Andrzej Stós (2013)

Studia Mathematica

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

Porosity of Collet–Eckmann Julia sets

Feliks Przytycki, Steffen Rohde (1998)

Fundamenta Mathematicae

We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.

Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien

Zhan Shi (2004/2005)

Séminaire Bourbaki

La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes...

Propriétés arithmétiques et dynamiques du fractal de Rauzy

Ali Messaoudi (1998)

Journal de théorie des nombres de Bordeaux

Dans ce travail, nous construisons explicitement deux isomorphismes métriques partout continus. L’un entre le système dynamique symbolique associé à la substitution σ : 0 01 , 1 02 , 2 0 et une rotation sur le tore 𝕋 2 ; l’autre, entre le système adique stationnaire [33] associé à la matrice de la substitution et la même rotation. Pour cela, nous étudions les propriétés arithmétiques de la frontière d’un ensemble compact de appelé “fractal de Rauzy”. Les constructions se généralisent aux substitutions de la forme σ k : 0 01 , 1 02 , k - 1 0 k , k 0 ...

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