$AH$-substitution and Markov Partition of a Group Automorphism on $T^d$

Abstract

The existence of a Markov partition of a hyperbolic group automorphism generated by an integral matrix with determinant $\pm 1$ is established by Sinai (see [22]). After that, there are many articles to construct Markov partitions of group automorphisms generated by non-negative matrices satisfying Pisot condition by the tiling method from substitutions (see [1], [7], [16], [19], [5]). One of the purpose of this paper is to establish the construction method of a Markov partition for a group automorphism generated by a non-positive matrix satisfying ``negative Pisot'' condition. An anti-homomorphic extension of a substitution, called $AH$-substitution, is introduced in the paper. Owing to this new substitution, the Markov partition of the group automorphism from the non-positive integral matrix is constructed.