Hausdorff dimension for randomly perturbed self affine attractors

Thomas Jordan, Mark Pollicott, K. Simon

Research output: Contribution to journalArticle

45 Citations (Scopus)

Abstract

In this paper we shall consider a self-affine iterated function system in ℝd, d ≥ 2, where we allow a small random translation at each application of the contractions. We compute the dimension of a typical attractor of the resulting random iterated function system, complementing a famous deterministic result of Falconer, which necessarily requires restrictions on the norms of the contraction. However, our result has the advantage that we do not need to impose any additional assumptions on the norms. This is of benefit in practical applications, where such perturbations would correspond to the effect of random noise. We also give analogous results for the dimension of ergodic measures (in terms of their Lyapunov dimension). Finally, we apply our method to a problem originating in the theory of fractal image compression.

Original languageEnglish
Pages (from-to)519-544
Number of pages26
JournalCommunications in Mathematical Physics
Volume270
Issue number2
DOIs
Publication statusPublished - Mar 2007

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Self-affine
Hausdorff Dimension
Attractor
Iterated Function System
norms
contraction
Contraction
Fractal Image Compression
Norm
Ergodic Measure
Random Systems
Affine Systems
Random Noise
random noise
Lyapunov
fractals
constrictions
Restriction
Perturbation
perturbation

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Hausdorff dimension for randomly perturbed self affine attractors. / Jordan, Thomas; Pollicott, Mark; Simon, K.

In: Communications in Mathematical Physics, Vol. 270, No. 2, 03.2007, p. 519-544.

Research output: Contribution to journalArticle

Jordan, Thomas ; Pollicott, Mark ; Simon, K. / Hausdorff dimension for randomly perturbed self affine attractors. In: Communications in Mathematical Physics. 2007 ; Vol. 270, No. 2. pp. 519-544.
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