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

Manning, Anthony

doi:10.1090/S0002-9947-02-03003-9

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