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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Orbits of paths under hyperbolic toral automorphisms


Author: Ricardo Mañé
Journal: Proc. Amer. Math. Soc. 73 (1979), 121-125
MSC: Primary 58F15; Secondary 58F10
MathSciNet review: 512072
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Abstract: A hyperbolic toral automorphism is a map $ f:{T^n} \hookleftarrow$ such that has a linear lifting $ L:{{\mathbf{R}}^n} \hookleftarrow$ without eigenvalues of modulus 1. In this note we prove that the orbit under f of a rectifiable nonconstant path $ \gamma :[a,b] \to {T^n}$ contains a coset of a toral subgroup invariant under same power of f. For $ {C^2}$ paths the same result was proved by J. Franks. For $ {C^0}$ arcs S.G. Hancock proved that it is false.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0512072-3
Article copyright: © Copyright 1979 American Mathematical Society