Transitivity of Heisenberg group extensions of hyperbolic systems

Melbourne, Ian; Niţică, Viorel; Török, Andrei

doi:10.1017/s014338571000091x

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…We next consider the case of skew products in higher dimension. In related work, K. Burns and A. Wilkinson studied stable ergodicity of rotation extensions and of more general group extensions over Anosov diffeomorphisms [BW99], and M. Field, I. Melbourne, V. Niţicȃ, and A. Török have analyzed group extensions over Axiom A systems, proving results on transitivity, ergodicity, and rates of mixing [FMT05,FMT07,MNT12].…”

Section: Higher Dimensionsmentioning

confidence: 99%

Ergodic components of partially hyperbolic systems

Hammerlindl¹

2017

Comment. Math. Helv.

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This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a C 1 lamination and gives results about the accessibility classes of non-volume-preserving systems.

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Section: Higher Dimensionsmentioning

confidence: 99%

Ergodic components of partially hyperbolic systems

Hammerlindl¹

2017

Comment. Math. Helv.

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“…The main result in [48], improving on a weaker result showing only genericity and proven in [49], is:…”

Section: Remarkmentioning

confidence: 99%

Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

Niţică

Török

2015

Axioms

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Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics.

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“…This approach cannot be applied to nilpotent Lie groups due to the lack of a large set of compact elements. In [11] it is shown that the conjecture holds generically for extensions with fibre the standard real Heisenberg group H n of dimension 2n + 1. Moreover, if one considers extensions with fibre 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%

“…In this paper, we prove the conjecture for extensions with fibre the standard real Heisenberg group H n . Our approach builds on the techniques developed in [8,10,11,15]. In particular, in [11] results from the classical theory of Diophantine approximation, such as Minkowski Diophantine result for indefinite quadratic forms, come for the first time into play.…”

Section: Introductionmentioning

confidence: 99%

Stable transitivity of Heisenberg group extensions of hyperbolic systems

Niţică

Török

2014

Nonlinearity

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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.

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Transitivity of Heisenberg group extensions of hyperbolic systems

Cited by 6 publications

References 13 publications

Ergodic components of partially hyperbolic systems

Ergodic components of partially hyperbolic systems

Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

Stable transitivity of Heisenberg group extensions of hyperbolic systems

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