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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-4,
author = {Bal\'azs B\'ar\'any},
title = {On the Hausdorff dimension of a family of self-similar sets with complicated overlaps},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {49-59},
zbl = {1194.28006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-4}
}
Balázs Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fundamenta Mathematicae, Tome 205 (2009) pp. 49-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-4/