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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Markov partition that reflects the geometry of a hyperbolic toral automorphism
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by Anthony Manning PDF
Trans. Amer. Math. Soc. 354 (2002), 2849-2863 Request permission

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
  • Received by editor(s): September 4, 2001
  • Published electronically: February 26, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2849-2863
  • MSC (2000): Primary 37D20, 37B10; Secondary 28A80, 37B40
  • DOI: https://doi.org/10.1090/S0002-9947-02-03003-9
  • MathSciNet review: 1895206