Infinite component Rauzy Fractal

Discovered by


An invertible substitution rule with a disconnected Rauzy Fractal. For two letter substitution rules the Rauzy fractal is connected if and only if the substitution is invertible. In fact as the window is one dimensional for these tilings it is an interval. It was hoped that the connectedness property extended to the higher dimensional case. Unfortunately, as this example shows, this is not the case. A second example, with just two components is 2-component Rauzy fractal.

For more details on Rauzy fractals and connectedness see for example Sie04.

The Rauzy fractal is: rauzy fractal

Substitution Rule

Rule Infinite component Rauzy Fractal


Patch Infinite component Rauzy Fractal


Siegel, Anne
Pure discrete spectrum dynamical system and periodic tiling associated with a substitution
Ann. Inst. Fourier (Grenoble) 2004, 54, 2, pp. 341--381,