Work

The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

Year: 2025

Type: article

Abstract: We provide two methods to characterise the connectedness of all d-dimensional generalised Sierpiński sponges whose corresponding iterated function systems (IFSs) are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in ${\mathbb {R}}^d$ . By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$ . Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

Source: Mathematical Proceedings of the Cambridge Philosophical Society

DOI: 10.1017/s0305004125000015

Authors: HUO-JUN RUAN, JIAN-CI XIAO

Institutions: Zhejiang University, Nanjing University of Aeronautics and Astronautics

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Chinese University of Hong Kong, Fundamental Research Funds for the Central Universities, Natural Science Foundation of Zhejiang Province