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.

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