Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385704000355How to cite this article: MICHAEL FIELD, IAN MELBOURNE and ANDREI TÖRÖK (2005). Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets.Abstract. We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C 2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the C r -topology for all r > 0 (here r need not be an integer and C 1 is replaced by Lipschitz). Moreover, when r ≥ 2, we show that there is a C 2 -open and C r -dense subset of C r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R m -extensions, thereby generalizing a result of Niţicȃ and Pollicott, and on stable mixing for suspension flows. THEOREM 1.7. For r > 0, there exists a C r -open and -dense subset W r of S r such that for all f ∈ W r , the skew extension f :Remark 1.8. If is a hyperbolic attractor, then we obtain the improved result that W r is C 0 -open in S r . Again, C α -openness, α ∈ [0, 1), fails for general hyperbolic basic sets.1.3. Suspension flows. We now describe our main result on the stability of weak mixing for suspensions flows over hyperbolic basic sets. Let R r denote the space of strictly positive functions (roof functions) in C r (M, R). Each roof function f ∈ R r defines a suspension flow f t : f → f on the suspension f . When = M is Anosov, it follows from the Anosov alternative [2] that f is mixing for all non-constant f ∈ R r .

Transitivity of Euclidean-type extensions of hyperbolic systems

“…By [2], the compact group extension f β G : X × G → X × G is transitive for an open dense set of C r cocycles β : X → Γ. In this section we show that we can specify the G-component of a particularβ j without significantly changing the remainingβ j s by modifying the heteroclinic cycle p 1 , .…”

“…By [2], the compact cocycle β G : X → G is transitive for an open dense set of C r cocycles β : X → Γ. Hence, we may suppose without loss that β G is stably transitive.…”

“…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).…”

Transitivity of Heisenberg group extensions of hyperbolic systems

“…A necessary condition for transitivity is that β ∈ S r (X, R d × K). First we recall a result of [10,2,7].…”

“…In [6] we proposed a general conjecture about transitivity: namely that modulo obstructions appearing from the fact that the range of the cocycle is included in a maximal semigroup with non-empty interior, the set of C r transitive cocycles contains an open and dense subset. The conjecture is proved for various classes of Lie groups, mostly semidirect products of compact and Euclidean, in [2,5,6,7,10]. An important test case is presented by the special Euclidean group Γ = SE(n) = SO(n) R n .…”