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 ). 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 , , , , ). 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.
"$AH$-substitution and Markov Partition of a Group Automorphism on $T^d$." Tokyo J. Math. 31 (2) 375 - 398, December 2008. https://doi.org/10.3836/tjm/1233844059