Instability for the rotation set of homeomorphisms of the torus homotopic to the identity

Abstract

In this paper we consider homeomorphisms $f:T^2 \rightarrow T^2$ homotopic to the identity and their rotation sets $\rho (\tilde{f})$, which are compact convex subsets of the plane. We show that if $\rho (\tilde{f})$ has an extremal point $(t,\omega )$ which is not a rational vector, then arbitrarily C⁰ close to f we can find a homeomorphism g such that $\rho (\tilde{g})\cap \rho (\tilde{f})^c\neq \emptyset$. So in this case, we have instability for the rotation set.