Gap sequences of fractal squares

Gap sequences and Topological properties of Bedford-McMullen sets

“…By Theorem 2.2, we immediately obtain the following result on fractal squares, which generalizes the main result of [20], where all (possible) nontrivial connected components of a fractal square are parallel line segments.…”

“…Currently, most work on gap sequences focuses on self-similar and self-conformal fractals with totally disconnected condition, see [3,4,28,34]. Recently, Liang and Ruan [20] studied the gap sequences of fractal squares (a special class of self-similar sets), where they allowed the sets containing connected components.…”

“…In [22], Miao, Xi and Xiong obtained the asymptotic estimate of gap sequences of totally disconnected Bedford-McMullen sets. In [20], Liang and Ruan studied the gap sequences of fractal squares, where all (possible) nontrivial connected components are parallel line segments. Note that fractal squares can be regarded as a special class of Bedford-McMullen sets, see Subsection 2.1 for definitions.…”

“…Note that fractal squares can be regarded as a special class of Bedford-McMullen sets, see Subsection 2.1 for definitions. Inspired by [20,22], we reveal the connection between topological properties and gap sequences and obtain the asymptotic estimate of gap sequences of Bedford-McMullen sets without any artificial restriction. For each w = (w (1) , w (2) ) ∈ D, we define a self-affine transformation on R 2 by (2.1)…”

“…Specially, for N = M, the Bedford-McMullen set E becomes a self-similar set, and we call it a fractal square, and denote it by F (N, D). See [18,20,29,30,32] for various studies on the typical fractals.…”