Topological properties of self-similar fractals with one parameter

Abstract

Abstract. In this paper, we study two classes of planar self-similar fractals T ε with a shifting parameter ε. The first one is a class of self-similar tiles by shifting x-coordinates of some digits. We give a detailed discussion on the disk-likeness (i.e., the property of being a topological disk) in terms of ε. We also prove that T ε determines a quasi-periodic tiling if and only if ε is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for T ε to be connected.

Thresholds for one-parameter families of affine iterated function systems*

“…If the dimension d = 2 and the eigenvalues of M are complex, then, via a conjugacy, the IFS F M is equivalent to the IFS F τ of (2.1). In this case, their result states that In [15,16] the authors study connectedness and disk-likeness of the attractor for a particular family of self-similar digit tiles (see [20]) in R 2 . The family depends on a parameter involving the digits.…”

Thresholds for One-Parameter Families of Affine Iterated Function Systems

“…Call an IFS F in R d degenerate if there exists an invariant affine subspace of dimension less than d that is common to all the functions in F ; otherwise call F non-degenerate. Hare and Siderov [12,13] In [15,16] the authors study connectedness and disk-likeness of the attractor for a particular family of self-similar digit tiles (see [20]) in R 2 . The family depends on a parameter involving the digits.…”

Fractal squares with finitely many connected components *

“…Another interesting topic is the Lipschitz equivalence between self-similar sets, for which one can refer to Falconer and Marsh [4], Luo and Lau [12], Rao-Ruan-Wang [16,17] etc. Research on other aspects can be seen in Bandt and Keller [1], Luo and Wang [14], Roinestad [19,20] etc.…”