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

Volume 206 / 2009

Balázs Bárány Fundamenta Mathematicae 206 (2009), 49-59 MSC: Primary 28A80; Secondary 37C45, 28A78. DOI: 10.4064/fm206-0-4


We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) $\left\{\gamma x,\lambda x, \lambda x+1\right\}$. 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 $(\gamma, \lambda), \gamma < \lambda$. This result only holds for almost all parameters: we find a dense set of parameters $(\gamma, \lambda)$ for which the Hausdorff dimension of the attractor is strictly smaller.


  • Balázs BárányDepartment of Stochastics
    Institute of Mathematics
    Technical University of Budapest
    P.O. Box 91
    1521 Budapest, Hungary

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