Stepped surfaces and Rauzy fractals induced from automorphisms on the free group of rank 2

From irreducible unimodular Pisot substitutions of rank 2, we can construct the discrete approximations of the lines, the so-called stepped surfaces, and moreover, we obtain Rauzy fractals as geometric realizations of the symbolic dynamical systems. In this paper, we study a particular infinite family of automorphisms on the free group of rank 2. We see that for every hyperbolic quadratic unit integer λ, there is a corresponding automorphism in the infinite family whose incidence matrix has eigenvalue λ; and the automorphism is conjugate to a substitution or “an alternating substitution”. And we construct stepped surfaces and Rauzy fractals induced from automorphisms in the family by using tiling substitutions. We also discuss the domain exchange transformations on Rauzy fractals.