Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real C ∞ -coboundaries of such a diffeomorphism is closed in C ∞ (T) if and only if its rotation number is Diophantine.
Abstract. We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré-Birkhoff Theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has nonempty interior.
We prove a Livšic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary.
A smooth vector field X on a closed orientable d-manifold M is said to be cohomologically rigid when given anywhere L X is the Lie derivative in the X direction. In 1984, Anatole Katok conjectured that every cohomologically rigid vector field should be smoothly conjugated to a Diophantine vector field on the d-torus T d . In this work the validity of the Katok conjecture for 3-manifolds is proved.
RésuméUn champ de vecteurs X sur une variété M compacte orientable de dimension d est dit cohomologiquement rigide si pour touteoù L X désigne la dérivée de Lie dans la direction de X. En 1984, Anatole Katok a conjecturé que tout champ de vecteurs cohomologiquement rigide devrait être conjugué par un difféomorphisme lisse à un champ linéaire diophantien sur le tore T d . Dans ce travail nous démontrons la conjecture de Katok pour les variétés de dimension trois.
Abstract. We consider diffeomorphisms in ∞ ( 2 ), the ∞ -closure of the conjugancy class of translations of 2 . By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-type property, which takes into account the "directions" of mixing, and we prove that generic elements of ∞ ( 2 ) satisfy this property.As a consequence, we obtain a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.
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