Fractal tiles and quasidisks (Q2512971)

A John domain is a simple connected region \(S\) in the plane such that every point can be reached from a fixed base point by a flexible cone in \(S\) with a fixed vertex angle. These regions were introduced in 1961 by \textit{F. John} [Commun. Pure Appl. Math. 14, 391--413 (1961; Zbl 0102.17404)]. This paper deals with John domains and quasidisks (linearly connected John domains). The authors prove that there exists a disklike planar self-affine tile (homeomorphic to a closed disk) whose interior is not a quasidisk. For this, they construct a counterexample to show that the interior of a self-affine tile need not to be a John domain or a linearly connected domain. They also prove that the interiors of many disklike self-similar tiles (including some classical ones) are quasidisks. It is also proved that adding a certain property the interior of any self-similar tile is necessarily a quasidisk (a John domain and a linearly connected domain).