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

摘要

We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {yx, Ax, Ax + 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 ('y, A), -y < A. This result only holds for almost all parameters: we find a dense set of parameters (-y, A) for which the Hausdorff dimension of the attractor is strictly smaller.