Orbits of paths under hyperbolic toral automorphisms

Abstract

A hyperbolic toral automorphism is a map f : T n ↩ f:{T^n} \hookleftarrow such that has a linear lifting L : R n ↩ 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 γ : [ a , b ] → T n \gamma :[a,b] \to {T^n} contains a coset of a toral subgroup invariant under same power of f. For C 2 {C^2} paths the same result was proved by J. Franks. For C 0 {C^0} arcs S.G. Hancock proved that it is false.