Algebraic numbers, free group automorphisms and substitutions on the plane

There has been much recent work on the geometric representation of Pisot substitutions and the explicit construction of Markov partitions for Pisot toral automorphims. We give a construction that extends this to the general hyperbolic case. For the sake of simplicity, we consider a simple example of an automorphism of the free group on 4 generators whose associated matrix has 4 distinct complex eigenvalues, two of them of modulus larger than 1 and the other 2 of modulus smaller than 1 (non-Pisot case). Using this generator, we build substitution polygonal tilings of the contracting plane and the expanding plane of the matrix. We prove that these substitution tilings can be lifted in a unique way to stepped surfaces approximating each of these planes. The vertices of each of these stepped surfaces can be projected to an "atomic surface'', a compact set with fractal boundary contained in the other plane. We prove that both tilings can be refined to exact self-similar tilings whose tiles have fractal boundaries and can be obtained by iteration or by a ``cut and project'' method by using the atomic surface as the window. Using the self-similar tiling, one can build a numeration system associated to a complex $ \lambda$-expansion; the natural extension of the $ \lambda$-expansion associated with this number system is the linear map obtained by abelianization of the free group automorphism. This gives an explicit Markov partition of this hyperbolic toral automorphism. The fractal domains can be used to define a pseudo-group of translations which gives transversal dynamics in the sense of Vershik (1994) or numeration systems in the sense of Kamae (2005). The construction can be extended to a larger class of free group automorphisms, each of which can be used to build substitution rules and dynamical systems.