The rotation set and periodic points for torus homeomorphisms

1996/05/07 by Franks, John
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/9605228

Abstract

We consider the rotation set ρ(F) for a lift F of an area preserving homeomorphism f: \t²→ \t², which is homotopic to the identity. The relationship between this set and the existence of periodic points for f is least well understood in the case when this set is a line segment. We show that in this case if a vector v lies in ρ(F) and has both co-ordinates rational, then there is a periodic point x∈ \t² with the property that \fracF^q(x₀)-x₀q = v where x₀∈ \re² is any lift of x and q is the least period of x.

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