ON SELF-SIMILARITY OF BOUNDED REGIONS IN THE PLANE

Wen posed the following question at the Chinese Conference on Fractal Geometry and Dynamical Systems 2023: under what conditions is a quadrangle a self-similar set with the open set condition? Xu, Xi and Jiang [On self-similarity of quadrangle, Fractals 32(5) (2024) 2450096] proved that for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the open set condition. Motivated by this result and Wen’s question, it is natural to consider the self-similarity of a bounded planar region with positive two-dimensional Lebesgue measure. In this paper, we partially address this problem as follows. Suppose [Formula: see text] is a [Formula: see text] non-degenerate Jordan curve in the plane such that any point on [Formula: see text] has non-zero curvature. Let [Formula: see text] be a bounded closed region with [Formula: see text] as its boundary. Then [Formula: see text] is not a self-similar set. Similar result can be obtained in [Formula: see text].