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A Markov partition that reflects the geometry of a hyperbolic toral automorphism
Author(s):
Anthony
Manning
Journal:
Trans. Amer. Math. Soc.
354
(2002),
2849-2863.
MSC (2000):
Primary 37D20, 37B10;
Secondary 28A80, 37B40
Posted:
February 26, 2002
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Abstract:
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.
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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:
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
Posted:
February 26, 2002
Copyright of article:
Copyright
2002,
American Mathematical Society
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