The exact Hausdorff dimension in random recursive constructions

Graf, Siegfried; Mauldin, R. Daniel; Williams, Stanley C.

doi:10.1090/memo/0381

Cited by 96 publications

(77 citation statements)

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“…It would be worth extending the present work to study random multifractal zeta functions, first in the same setting as [10], and later on, in the broader framework of random fractals and multifractals considered, for example, in [1,8,9,14,35,41,42].…”

Section: Concluding Commentsmentioning

confidence: 99%

Fractal Strings and Multifractal Zeta Functions

2009

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Abstract. For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

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Section: Concluding Commentsmentioning

confidence: 99%

Fractal Strings and Multifractal Zeta Functions

2009

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“…Since t 0 = t 1 = 1, Lemma 2.6 of [25] yields lim sup k→∞ t 1 k k < ∞. This implies the existence of a constant C > 0 such that…”

Section: Proof Of Theorem 35mentioning

confidence: 91%

Renewal of singularity sets of random self-similar measures

Barral

Seuret

2007

Adv. Appl. Probab.

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“…When the scheme incorporates a randomised step, then the ensuing set may be termed a`random fractal'. Such sets may be studied in some generality (see 131,153,183,313]), and properties of fractal dimension may be established. The following simple example is directed at a`percolative' property, namely the possible existence in the random fractal of long paths.…”

Section: General Labyrinthsmentioning

confidence: 99%

Percolation and disordered systems

Grimmett

1997

Lecture Notes in Mathematics

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PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematical theory of percolation and related topics. The rst nine chapters summarise rigorous results in percolation theory, with special emphasis on results obtained since the publication of my book 156] entitled`Percolation', and sometimes referred to simply as G] in these notes. Following this core material are chapters on random walks in random labyrinths, and fractal percolation. The nal two chapters include material on Ising, Potts, and random-cluster models, and concentrate on a`percolative' approach to the associated phase transitions.The rst target of this course is to draw a picture of the mathematics of percolation, together with its immediate mathematical relations. Another target is to present and summarise recent progress. There is a considerable overlap between the rst nine chapters and the contents of the principal reference G]. On the other hand, the current notes are more concise than G], and include some important extensions, such as material concerning strict inequalities for critical probabilities, the uniqueness of the in nite cluster, the triangle condition and lace expansion in high dimensions, together with material concerning percolation in slabs, and conformal invariance in two dimensions. The present account di ers from that of G] in numerous minor ways also. It does not claim to be comprehensive. A second edition of G] is planned, containing further material based in part on the current notes.A special feature is the bibliography, which is a fairly full list of papers published in recent years on percolation and related mathematical phenomena. The compilation of the list was greatly facilitated by the kind responses of many individuals to my request for lists of publications.Many people have commented on versions of these notes, the bulk of which have been typed so superbly by Sarah Shea-Simonds. I thank all those who have contributed, and acknowledge particularly the suggestions of Ken Alexander, Carol Bezuidenhout, Philipp Hiemer, Anthony Quas, and Alan Stacey, some of whom are mentioned at appropriate points in the text. In addition, these notes have bene ted from the critical observations of various members of the audience at St Flour.Members of the 1996 summer school were treated to a guided tour of the library of the former seminary of St Flour. We were pleased to nd there a copy of the Encyclop edie, ou Dictionnaire Raisonn e des Sciences, des Arts et des M etiers, compiled by Diderot and D'Alembert, and published in Geneva around 1778. Of the many illuminating entries in this substantial work, the following de nition of a probabilist was not overlooked. 3 PROBABILISTE, s. m. (Gram. Th eol.) celui qui tient pour la doctrine abominable des opinions rendues probables par la d ecision d'un casuiste, & qui assure l'innocence de l'action faite en cons equence. Pascal a foudroy e ce systême, qui ouvroit la porte au crime en accordant a l'autorit e les pr erogatives de la certitude, a l'opinion & la s ec...

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The exact Hausdorff dimension in random recursive constructions

Cited by 96 publications

References 1 publication

Fractal Strings and Multifractal Zeta Functions

Fractal Strings and Multifractal Zeta Functions

Renewal of singularity sets of random self-similar measures

Percolation and disordered systems

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