Families of connected self-similar sets generated by complex trees

Abstract

The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain oneparameter families of connected self-similar sets F A (z). In order to study topological changes of F A (z) in regions R ⊆ C where these families are defined, we introduce a new kind of set M ⊆ R which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set M 0 ⊂ M related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements F A (z) disconnected for parameters z ∈ R\M. Keywords complex trees • self-similar sets • IFSs • Hausdorff dimension • OSC • fractal dendrites • fractal trees