Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

2011

Ergod. Th. Dynam. Sys.

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We show that among C r extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group H n of dimension 2n + 1, those that avoid an obvious obstruction to topological transitivity are generically topological transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with compact commutator subgroup (for example, H n /Z), among those that avoid the obvious obstruction, topological transitivity is open and dense.

Section: Introductionmentioning

confidence: 99%

Transitivity of Heisenberg group extensions of hyperbolic systems

Melbourne¹,

2011

Ergod. Th. Dynam. Sys.

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“…In [5] we proposed a general conjecture about transitivity in the class of Hölder cocycles: namely that modulo obstructions appearing from the fact that the range of the cocycle is included in a semigroup, transitivity is open and dense. The conjecture is proved for various classes of Lie groups, mostly semidirect products of compact and Euclidean, in [2,4,5,6,8]. In addition [5] exhibits open sets of C r transitive cocycles with fiber Sp(n).…”

Section: Introductionmentioning

confidence: 99%

“…The conjecture is proved for various classes of Lie groups, mostly semidirect products of compact and Euclidean, in [2,4,5,6,8]. In addition [5] exhibits open sets of C r transitive cocycles with fiber Sp(n). An important test case is presented by cocycles with fiber the special Euclidean group SE(n) = SO(n) R n .…”

Section: Introductionmentioning

confidence: 99%

“…An important test case is presented by cocycles with fiber the special Euclidean group SE(n) = SO(n) R n . It is shown in [4,5,6] that when n is even the set of cocycles that are transitive is Hölder-open and C r -dense. The conjecture remains unsolved for n ≥ 3 odd.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we recall some general results from [5]. In Section 3, we introduce a new construction that seems particularly useful for cocycles that satisfy a subexponential growth condition.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Transitivity of Euclidean-type extensions of hyperbolic systems

Melbourne

2009

Ergod. Th. Dynam. Sys.

Abstract. Let f : X → X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of C r , r > 0, cocycles with fiber special Euclidean group SE(n) those that are transitive form a residual set (countable intersection of open dense sets). This result is new for n ≥ 3 odd.More generally, we consider Euclidean-type groups G R n where G is a compact connected Lie group acting linearly on R n . When Fix G = {0}, it is again the case that the transitive cocycles are residual. When Fix G = {0}, the same result holds on restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.

Stable transitivity of Heisenberg group extensions of hyperbolic systems

2014

Nonlinearity

Self Cite

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.