Radix expansions and connectedness of planar self-affine fractals (Q2006798)

Let \(A\) be an expanding \(2\times 2\) matrix whose characteristic polynomial is of the form \(f(x) = x^2+px+3\). For \(k, \ell\in \mathbb{Z}\) and \(v\in \mathbb{R}^2\), let \(\mathcal{D} := \{0, v, \ell v + k A v\}\) denote a digit set. Assume that \(\{v, Av\}\) is linearly independent. The theory of iterated function systems yields the existence of a unique self-affine fractal \(T\) satisfying \(AT = T + \mathcal{D}\). Using radix expansions, the authors provide a complete characterization (in terms of \(p\), \(k\), and \(\ell\)) in case that \(T\) is connected.