The dimension theory of almost self-affine sets and measures

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Abstract

A self-affine IFS F = {fi (x) = Aix + ti}ii=1 is a finite list of contracting affine maps on ℝd, for some d ≥ 1. The attractor of F is (Formula presented.) where B is a sufficiently large ball centered at the origin. In most cases we cannot compute the dimension of f. However, if we add an independent additive random error to each fik in (0.1) then the dimension of this random perturbation (called almost self-affine system) is almost surely the so-called affinity dimension of theoriginal deterministic system. The dimension theory of almost self-affine sets and measures were described in Jordan et al. (Commun. Math. Phys. 270(2):519-544, 2007). The multifractal analysis of almost self-affine measures has been studied insome recent papers (Falconer, Nonlinearity 23:1047-1069, 2010; Barral and Feng, Commun. Math. Phys. 318(2):473-504, 2013). In the second part of this note I give a survey of this field but first we review some results related to the dimension theory of self-affine sets

Original languageEnglish
Title of host publicationFractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets
EditorsV. Kannan, Michael F. Barnsley, Robert L. Devaney, Vinod Kumar P.B., Kenneth J. Devaney, Christoph Bandt
PublisherSpringer New York LLC
Pages103-127
Number of pages25
ISBN (Electronic)9783319081045
DOIs
Publication statusPublished - Jan 1 2014
Event1st International Conference and Workshop on Fractals and Wavelets, ICFW India - Kochi, India
Duration: Nov 13 2013Nov 16 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume92
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other1st International Conference and Workshop on Fractals and Wavelets, ICFW India
CountryIndia
CityKochi
Period11/13/1311/16/13

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Keywords

  • Hausdorff dimension
  • Processes in random environment
  • Random fractals

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Simon, K. (2014). The dimension theory of almost self-affine sets and measures. In V. Kannan, M. F. Barnsley, R. L. Devaney, V. K. P.B., K. J. Devaney, & C. Bandt (Eds.), Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets (pp. 103-127). (Springer Proceedings in Mathematics and Statistics; Vol. 92). Springer New York LLC. https://doi.org/10.1007/978-3-319-08105-2__6