Orbits of paths under hyperbolic toral automorphisms

Mañé, Ricardo

doi:10.1090/S0002-9939-1979-0512072-3

Orbits of paths under hyperbolic toral automorphisms
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Proc. Amer. Math. Soc. 73 (1979), 121-125 Request permission

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.

References

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Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors