On a family of self-affine sets: topology, uniqueness, simultaneous expansions

TL;DR

This paper analyzes the topological and measure-theoretic properties of a family of self-affine sets defined by two parameters, revealing new bounds for interior points, connectedness, and unique expansions, advancing understanding of their structure.

Contribution

It provides new bounds for when the self-affine set has non-empty interior and explores the connectedness and uniqueness properties of expansions in this family.

Findings

If $eta_1<eta_2<1.202$, then $A$ has a non-empty interior.

The connectedness locus for the family is not simply connected.

The set of points with a unique address has positive Hausdorff dimension.

Abstract

Let $\beta_1,\beta_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{\beta_1}, \frac{y+i}{\beta_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{\beta_1, \beta_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a detailed analysis of $A$, and the parameters $(\beta_1, \beta_2)$ where$A$ satisfies various topological properties. In particular, we show that if $\beta_1<\beta_2<1.202$,then $A$ has a non-empty interior, thus significantly improving the bound from [1]. In the opposite direction,we prove that the connectedness locus for this family studied in [16] is not simply connected.We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(\beta_1,\beta_2)$.Finally, we investigate simultaneous $(\beta_1,\beta_2)$-expansions of reals, which were the initial motivation for studying this family in [5].