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Common homoclinic points of commuting toral automorphisms

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Abstract

The points homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a ℤ2-action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same homoclinic points or have no homoclinic point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    Anthony Manning

  2. Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090, Vienna, Austria

    Klaus Schmidt

  3. Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090, Vienna, Austria

    Klaus Schmidt

Authors
  1. Anthony Manning
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  2. Klaus Schmidt
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Correspondence to Anthony Manning.

Additional information

The second author would like to thank the Austrian Academy of Sciences and the Royal Society for partial support while this work was done.

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Manning, A., Schmidt, K. Common homoclinic points of commuting toral automorphisms. Isr. J. Math. 114, 289–299 (1999). https://doi.org/10.1007/BF02785584

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  • DOI: https://doi.org/10.1007/BF02785584

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