A Markov partition that reflects the geometry of a hyperbolic toral automorphism

We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the -torus. The transition matrix is the transpose of the matrix induced by the automorphism in -dimensional homology, provided this is non-negative. (Here denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The rectangles are constructed by an iterated function system, and they resemble the product of the projection of a -dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal -dimensional face onto the stable subspace.