Structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points

We study the structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points. This property generalizes the local expansivity property, and at present there are no known examples of minimal C ² actions of finitely generated groups of circle diffeomorphisms for which this generalized property does not hold.     

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms ofT ⁿ which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservativeC ¹ perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [4] and so we answer the question posed there on the stable ergodicity of such systems.     

Robust rigidity for circle diffeomorphisms with singularities

We prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities, their C ¹-smooth equivalence follows from exponential convergence of those renormalizations. As an easy corollary, any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are C ¹-smoothly conjugate.     

The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Smooth partitions of Anosov diffeomorphisms are weak Bernoulli

It is shown that smooth partitions are weak Bernoulli forC ² measure preserving Anosov diffeomorphisms. A related type of coding is defined and an invariant discussed. Supported by the Sloan Foundation and NSF GP-14519.     

Examples of C1-smoothly conjugate diffeomorphisms of the circle with break that are not C1+γ-smoothly conjugate

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C ^1+γ -smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C ¹-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.     

On the Uniqueness of Hofer��s Geometry

We study the class of pseudo-norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such pseudo-norm that is continuous with respect to the C ^∞-topology, is dominated from above by the L _∞-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant pseudonorm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer’s metric.     

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

Actions de groupes de Kazhdan sur le cercle

We prove that every homomorphism from a discrete Kazhdan group to the group of orientation-preserving diffeomorphisms of the circle of class C^1+α (α>1/2) has a finite image.     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Conditions for a diffeomorghism to be embedded in aC r flow

We will consider aC ^r diffeomorphism of the real lineR, and give a necessary and sufficient condition for aC ^r diffeomorphism ofR to be embedded (uniquely) in aC ^r flow. As an application, we do the same for diffeomorphisms of the circleS ¹ and a class of analytic diffeomorphisms of the planeR ².     

Unique ergodicity of flows on the torus and the rotation number

We prove in this paper that any flow on the 2- torus with no singular points and periodic orbits which is generated by a vector fieldV=(P, Q) satisfyingV∈C ¹ orV∈C ⁰ andP≠0 is uniquely ergodic. Then we give an expression of the rotation number by using an invariant measure of a flow.     

Diffeomorphisms with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-shadowing property

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L ^p -shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C ¹-interior of the set of diffeomorphisms having L ^p -shadowing property.     

Récurrence et généricité

Résumé. Nous montrons un lemme de connexion C¹ pour les pseudo-orbites des difféomorphismes des variétés compactes. Nous explorons alors les conséquences pour les difféomorphismes C¹-génériques. Par exemple, les difféomorphismes conservatifs C¹-génériques (dune variété connexe) sont transitifs.

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We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms (on connected manifolds) are transitive.

     

The <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> generic diffeomorphism has trivial centralizer

Answering a question of Smale, we prove that the space of C ¹ diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.     

Herman’s theory revisited

We prove that a C ^2+α -smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D _δ , 0≤δ<α≤1, α−δ≠1, is C ^1+α−δ -smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms.     

Growth of Groups and Diffeomorphisms of the Interval

We prove that, for all α > 0, every finitely generated group of C ^1+α diffeomorphisms of the interval with sub-exponential growth is almost nilpotent. Consequently, there is no group of C ^1+α interval diffeomorphisms having intermediate growth. In addition, we show that the C ^1+α regularity hypothesis for this assertion is essential by giving a C ¹ counter-example. Received: November 2006, Revised: July 2007, Accepted: July 2007