Fractal tiles induced by tent maps

Abstract

In the present article, we deal with geometrical objects induced by the tent maps associated with special Pisot numbers that we call tent-tiles. They are compact subsets of the one-, two-, or three-dimensional Euclidean space, depending on the particular special Pisot number. Most of the tent-tiles have a fractal shape and we study the Hausdorff dimension of their boundary. Furthermore, we are concerned with tilings induced by tent-tiles. It turns out that tent-tiles give rise to two types of lattice tilings. In order to obtain these results we establish and exploit connections between tent-tiles and Rauzy fractals induced by substitutions and automorphisms of the free group.