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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Anthony Manning
Journal: Trans. Amer. Math. Soc. 354 (2002), 2849-2863
MSC (2000): Primary 37D20, 37B10; Secondary 28A80, 37B40
Published electronically: February 26, 2002
MathSciNet review: 1895206
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Abstract: We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the $n$-torus. The transition matrix is the transpose of the matrix induced by the automorphism in $u$-dimensional homology, provided this is non-negative. (Here $u$ 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 $(^n_u)$ rectangles are constructed by an iterated function system, and they resemble the product of the projection of a $u$-dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal $(n-u)$-dimensional face onto the stable subspace.


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Additional Information

Anthony Manning
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Email: akm@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03003-9
PII: S 0002-9947(02)03003-9
Keywords: Markov partition, hyperbolic toral automorphism, iterated function system
Received by editor(s): September 4, 2001
Published electronically: February 26, 2002
Article copyright: © Copyright 2002 American Mathematical Society