## On a family of two-dimensional self-aﬃne sets

#### Nikita Sidorov (University of Manchester)

Frank Adams Room 1, Alan Turing Building,

Let $$\beta_1,\beta_2>1$$ and $$T_i(x,y) = \big( \tfrac{x+i}{\beta_1},\tfrac{x+i}{\beta_2}\big), \, i \in \{\pm 1\}$$. Let $$A := A_{\beta_1, \beta_2}$$ be the unique non-empty compact set satisfying $$A = T_1(A) \cup T_{-1}(A)$$, i.e., the attractor of this IFS.

In this talk I will provide a detailed analysis of $$A$$, and the paramters $$(\beta_1, \beta_2)$$ where $$A$$ satisfies various topological properties, such as a non-empty interior for "small" $$\beta_1, \beta_2$$, and the connectedness locus for this family.  I will also talk about the set of uniqueness and simultaneous $$\beta$$-expansions.

This talk is based on joint work with Kevin Hare.