On the Hausdorff dimension of a family of self-similar sets with complicated overlaps
Volume 206 / 2009
Fundamenta Mathematicae 206 (2009), 49-59
MSC: Primary 28A80; Secondary 37C45, 28A78.
DOI: 10.4064/fm206-0-4
Abstract
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.
