On a Topological Problem of Strange Attractors

TL;DR

This paper investigates the connectedness of self-affine attractors generated by integer expanding matrices and finite digit sets, focusing on the unresolved cases in one and two dimensions.

Contribution

It provides new characterizations of connectedness for certain basic self-affine attractors in low dimensions, addressing gaps in existing research.

Findings

Connectedness criteria for specific attractors in one dimension.

Partial results on connectedness in two-dimensional cases.

Identification of open problems in the topology of self-affine sets.

Abstract

Somehow, the revised version of our paper \cite{KY} does not appear on journals' home page. Here we present the revised version altered to reflect the corrections and/or additions to that paper. In this note, we consider self-affine attractors that are generated by an integer expanding $n\times n$ matrix (i.e., all of its eigenvalues have moduli $>1$) and a finite set of vectors in ${\Bbb{Z}}^n$. We concentrate on the problem of connectedness for $n\leq 2$. Although, there has been intensive study on the topic recently, this problem is not settled even in the one-dimensional case. We focus on some basic attractors, which have not been studied fully, and characterize connectedness.