Tokyo Journal of Mathematics

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

Fumihiko ENOMOTO

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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.

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Tokyo J. Math., Volume 31, Number 2 (2008), 375-398.

First available in Project Euclid: 5 February 2009

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ENOMOTO, Fumihiko. $AH$-substitution and Markov Partition of a Group Automorphism on $T^d$. Tokyo J. Math. 31 (2008), no. 2, 375--398. doi:10.3836/tjm/1233844059.

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