On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group Diff_G^∞(M)₀ of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of Diff_G^∞(M)₀ when M is a G-manifold with codimension one orbit.     

On suboperators with codimension one domains

The question of the existence of a proper closed densely defined suboperator whose domain contains a given linear space is answered. Such suboperators with codimension one domains are described. The motivation for studying this topic comes from the theory of extremal extensions of (non-densely defined) positive operators.     

On codimension one foliations with prescribed cuspidal separatrix

In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. As an application, we will explain how this form could be used in order to study the analytic classification of the singularities via the projective holonomy, in the generalized surface case. We will also give an application to the analytic classification of singularities, and a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type.     

On codimension one foliations with Morse singularities on three-manifolds

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré-Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C^∞ foliation in the closed ball , with only one singularity which is of saddle type 2-2 and transverse to the boundary S³=∂B⁴.     

On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.      

On codimension one submersions of euclidean spaces

Summary It is shown that when (n–1) first integrals of a dynamical systemX inR ⁿ are known (and they are independent at any point ofR ⁿ ) then one can have (n3) that certain orbits ofX are diffeomorphic to circles and others diffeomorphic to straight-lines. An analytical criterion is also given (involving only the first derivatives of the first integrals) in order thatall the orbits be diffeomorphic to straight lines. Therefore the criterion is sufficient in order to avoid the presence of geometrical chaos among the orbits of the dynamical systemX.     

Anosov and expanding attractors

Science China Mathematics - An earlier conjecture is settled with an immersion of a 2-dimensional branched manifold. Possible obstructions in linear algebra and tiling theory are studied first.     

High dimension diffeomorphisms exhibiting infinitely many strange attractors

In this work we show, on a manifold of any dimension, that arbitrarily near any smooth diffeomorphism with a homoclinic tangency associated to a sectionally dissipative fixed or periodic point (i.e. the product of any pair of eigenvalues has norm less than 1), there exists a diffeomorphism exhibiting infinitely many Hénon-like strange attractors. In the two-dimensional case this has been proved in [E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 539–579]. We also show that a parametric version of this result is true.     

Classification of one-dimensional expanding attractors

It is shown that, in any dimension d ≥ 3, there exist diffeomorphisms of compact d-manifolds with one-dimensional expanding attractors which are conjugate on these attractors but not conjugate on their neighborhoods.     

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ (p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.     

On maximal ideals of codimension one in m-convex algebras

In Studia Math. 58:291–298, 1976, W. Żelazko gives a characterization of complex commutative complete unital m-convex algebras in which all maximal ideals are of codimension one. In these algebras every element has a bounded spectrum.

Here we present a similar result concerning sufficient conditions so that all maximal ideals of a complex commutative unital m-convex algebra to be of codimension one. Moreover, these conditions are proved to be equivalent to each other. We also give an example of a complex commutative unital advertibly complete m-convex algebra such that all its maximal ideals are of codimension one and has an element with unbounded spectrum.

     

On C1-class local diffeomorphisms whose periodic points are nonuniformly expanding

Using a sifting-shadowing combination, we prove in this paper that an arbitrary C¹-class local diffeomorphism f of a closed manifold M ⁿ is uniformly expanding on the closure ${\mathrm{Cl}_{M^n}(\mathrm{Per}(f))}$ of its periodic point set Per(f), if it is nonuniformly expanding on Per(f).     

Demushkin's Theorem in codimension one

Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n–1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X. Mathematics Subject Classification: 13A50, 14L30, 14M25, 14R20.     

Existence of codimension one foliations with minimal leaves

Every smooth closed manifold of dimension 4 or greater that has a smooth codimension one foliation, has such aC ¹ foliation whose leaves are minimal hypersurfaces for someC ¹ Riemannian metric.