Stable transitivity of Heisenberg group extensions of hyperbolic systems

Abstract

We consider skew-extensions with fibre the standard real Heisenberg group H n of a uniformly hyperbolic dynamical system.We show that among the C r 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 H n -extension is transitive if and only if the R 2n -extension given by the Abelianization of H n 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 S ⊂ H n , which avoids any maximal semigroup with non-empty interior, the closure of the semigroup generated by S is actually a group.

On a Semigroup Problem II

“…If X = R n is endowed with the Euclidean topology, then the semigroup problem has an affirmative answer [1,2]. A similar problem, in which separation by linear functionals is replaced by separation by certain maximal semigroups, was investigated for several classes of finite dimensional non-compact Lie groups such as Euclidean groups [1], nilpotent groups [3], and solvable groups [4].…”

Semigroup Conjectures for Central Semidirect Product of R^n with R^m

“…One can find in [6] a general conjecture about transitivity of such extensions: modulo the obstruction that the range of the cocycle is included in a maximal subsemigroup with nonempty interior, the set of C r transitive cocycles contains an open and dense subset of C r cocycles. The conjecture is proved for various classes of Lie groups G that are semidirect products of compact and abelian/nilpotent groups in [6,7,8,9,10,11,12].…”

“…The problem was solved for G = R n [11, Lemma 5] and more generally G = K × R n where K is a compact Lie group [6,Theorem 5.10]. It is also solved for G = SE(n) [6,Theorem 6.8] and for Heisenberg groups in [12,Theorem 8.6].…”