Dynamical properties of 2-torus parabolic maps

Abstract

In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations.