Stepped surfaces generated by automorphisms of the free groups

Hiromi Ei

Chuo University, Japan

When a Pisot, unimoduler substitution is given, one can consider Rauzy fractals and a stepped surface which derives a quasi periodic tiling of a plane. In recent researches, we begin to discuss the case without a real Pisot condition. In such case, automorphisms are used instead of substitutions. In the paper [1], the method to construct Rauzy fractals associated with an automorphism with some condition is proposed, using a double substitution. For example, the double substitution $\beta$ for the automorphism of rank 2 given by $\sigma(1)= 12^{-1}1, \sigma(2)= 12^{-1}$ is $\beta(1)= 1 \overline{2} 1,  \beta(2)= 1 \overline{2}, \beta(\overline{1})= \overline{1} 2 \overline{1},  \beta(\overline{2})= 2 \overline{1}$ over the alphabet ${\cal B}= \{1,2, \overline{1}, \overline{2} \}$. In my talk, I introduce the way to construct Rauzy fractals and a stepped surface associated with an automorphism without using a double substitution through some examples.

Pierre Arnoux, Valérie Berthé, Arnaud Hilion, and Anne Siegel, Fractal representation of the attractive lamination of an automorphism of the free group, Preprint, 2006.

Hiromi Ei, Some properties of invertible substitutions of rank $d$, and higher dimensional substitutions, OSAKA J. Math., vol. 40., 543-562, 2003.

Maki Furukado and Shunji Ito, Tilings generated by cubic complex Pisot companion matrices, Talk at Kanazawa, 2007.