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

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Manning, Anthony. (2002) A Markov partition that reflects the geometry of a hyperbolic toral automorphism. American Mathematical Society. Transactions, Vol.354 (No.7). pp. 2849-2863. ISSN 0002-9947

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Official URL: http://dx.doi.org/10.1090/S0002-9947-02-03003-9

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 (nu) 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Markov processes, Automorphisms, Torus (Geometry)
Journal or Publication Title:	American Mathematical Society. Transactions
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	2002
Volume:	Vol.354
Number:	No.7
Page Range:	pp. 2849-2863
Identification Number:	10.1090/S0002-9947-02-03003-9
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/4295