Geometry of symplectic partially hyperbolic automorphisms on 4-torus

Abstract

We study topological properties of automorphisms of 4-dimensional torus generatedby integer matrices being symplectic either with respect to the standard symplecticstructure in R4 or w.r.t. a nonstandard symplectic structure generated by an integer skew-symmetric nondegenerate matrix. Such symplectic matrix generates a partiallyhyperbolic automorphism of the torus, if its eigenvalues are a pair of reals outsidethe unit circle and a complex conjugate pair on the unit circle. The main classifying element is the topology of a foliation generated by unstable (stable) leaves ofthe automorphism. There are two different cases, transitive and decomposable ones.For the first case the foliation into unstable (stable) leaves is transitive, for the second case the foliation itself has a sub-foliation into 2-dimensional tori. For both cases the classification is given.