On the Hausdorff dimension of a family of self-similar sets with complicated overlaps

Balázs Bárány

Fundamenta Mathematicae (2009)

  • Volume: 206, Issue: 1, page 49-59
  • ISSN: 0016-2736

Abstract

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We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension of the attractor is strictly smaller.

How to cite

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Balázs Bárány. "On the Hausdorff dimension of a family of self-similar sets with complicated overlaps." Fundamenta Mathematicae 206.1 (2009): 49-59. <http://eudml.org/doc/282649>.

@article{BalázsBárány2009,
abstract = {We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) \{γx,λx,λx+1\}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension of the attractor is strictly smaller.},
author = {Balázs Bárány},
journal = {Fundamenta Mathematicae},
keywords = {Hausdorff dimension; iterated function system; transversality},
language = {eng},
number = {1},
pages = {49-59},
title = {On the Hausdorff dimension of a family of self-similar sets with complicated overlaps},
url = {http://eudml.org/doc/282649},
volume = {206},
year = {2009},
}

TY - JOUR
AU - Balázs Bárány
TI - On the Hausdorff dimension of a family of self-similar sets with complicated overlaps
JO - Fundamenta Mathematicae
PY - 2009
VL - 206
IS - 1
SP - 49
EP - 59
AB - We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension of the attractor is strictly smaller.
LA - eng
KW - Hausdorff dimension; iterated function system; transversality
UR - http://eudml.org/doc/282649
ER -

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