Families of connected self-similar sets generated by complex trees

2019/02/28 by Espigule, Bernat
#26C10 (Secondary) #28A78 #28A80 (Primary) 37F20 #37F45 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.11282

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 one-parameter 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\subseteqR which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set M₀\subseteqM 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\inR\backslashM.

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