Leírás
ELTE Analízis tanszék szemináruma
Abstract: We are given a finite list of contractioning similarity mappings $\{S_1,\dots, S_m\}$ on the real line. We call it a self-similar Iterated Function System (IFS). The attractor (the set that remains if we iterated this system on a sufficiently large interval infinitely many times) of self-similar IFSs are the (deterministic) self-similar sets. An interesting open problem is as follows: is there a self-similar set of positive Lebesgue measure and empty interior on the line? We consider this problem for randomly perturbed self-similar sets on the line. Such a set can be obtained as follows:
We start from a sufficiently large interval $I$. We obtain an $n$-cylinder $I_{ i_1,\dots,i_n }$ corresponding to the indices $(i_1,\dots,i_n)\in\{1,\dots, m\}^n$ of the (deterministic) self-similar set as follows: $ I_{ i_1,\dots,i_n } =S_{i_1}\circ\dots\circ S_{i_n}(I)$. In the randomly perturbed case, the corresponding cylinder is obtained by replacing $S_{i_k}$ by a random and independent of-everything translation of $S_{i_k}$ for all $k=1,\dots, n$. Then, from these randomly perturbed $n$-cylinders, we construct the randomly perturbed self-similar set exactly as we do in the deterministic case.
First, I review results related to the Lebesgue measure and Hausdorff dimension of these randomly perturbed self-similar sets. Then, I turn to our new result (joint with M. Dekking, B. Szekely, and N. Szekeres) about the existence of interior points in these randomly perturbed self-similar sets.