Stable transitivity of Heisenberg group extensions of hyperbolic systems

Abstract

We consider skew-extensions with fibre the standard real Heisenberg group of a uniformly hyperbolic dynamical system.We show that among the Cr extensions (r > 0) that avoid an obvious obstruction, those that are topologically transitive contain an open and dense set. More precisely, we show that an -extension is transitive if and only if the -extension given by the Abelianization of is transitive.A new technical tool introduced in the paper, which is of independent interest, is a diophantine approximation result. We show, under general conditions, the existence of an infinite set of approximate positive integer solutions for a diophantine system of equations consisting of a quadratic indefinite form and several linear equations. The set of approximate solutions can be chosen to point in a certain direction. The direction can be chosen from a residual subset of full measure of the set of real directions solving the system of equations exactly.Another contribution of the paper, which is used in the proof of the main result, but it is also of independent interest, is the solution of the so-called semigroup problem for the Heisenberg group. We show that for a subset , which avoids any maximal semigroup with non-empty interior, the closure of the semigroup generated by S is actually a group.