Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

The structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C ¹-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C ¹-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.     

Generic Bowen-expansive flows

We prove that in the set of all C ¹ vector fields on a compact manifold there is a residual subset which satisfies the property that if a vector field is Bowen-expansive, then it is Axiom A without cycles.     

Maximal transitive sets with singularities for genericC 1 vector fields

A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC ¹ generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC ¹ vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.     

Divergence-free vector fields with average and asymptotic average shadowing property

Let X be a divergence-free vector field on a three-dimensional compact connected Riemannian manifold. In this paper, we show that if X is in the C¹-interior of the set of divergence-free vector fields which satisfy the average shadowing property then X is Anosov. We also obtain similar result for asymptotic average shadowing property.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

Volume preserving flows with cyclic winding numbers groups and without periodic orbits on 3-manifolds

We construct examples of volume preserving non-singular C ¹ vector fields on closed orientable 3-manifolds, which have cyclic winding numbers groups with respect to the preserved volume element, but have no periodic orbits. Received: 17 January 1998 / Revised version: 31 March 1998     

On the Structure of the Solution Set of u″ = f(t,u), u(0) = u(1) = 0

The solution set of a Dirichlet problem x″ = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorphic to a closed subset of E. We prove that to an closed subset C of E there is a function f with Lipschitz constant arbitrarily close to π², such that the solution set of the corresponding Dirichlet problem is homeomorphic to C.     

Structural stability of vector fields with shadowing

Let X be a C¹ vector field without singularities. In this paper, we show that X is in the C¹ interior of the set of vector fields with the shadowing property if and only if X satisfies both Axiom A and the strong transversality condition; that is, X is structurally stable.     

Badly approximable vectors on fractals

For a large class of closed subsetsC of ℝ ⁿ , we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.     

Non-hyperbolic surfaces having all ideal triangles with finite area

We construct examples ofC ³ compact surfaces of non-positive curvature having non-Anosov geodesic flows and satisfying the following property: there existsL>0 such that the area of every ideal triangle in the universal covering of the surface is bounded above byL.Partially supported by CNPq of Brazilian Government     

Eine Charakterisierung der Sphäre im E3

It is well known, that in E³ the spheres are the closed convex C² -surfaces having the property, that each of their closed C²-curves with the total geodesic curvature O bisects the area of the surface. This characterization will be transmitted into the theory of convex surfaces founded by A.D.Alexandrov, where convex surfaces without any differentiability property are studied.     

The set of axiom A diffeomorphisms with no cycles

LetM be aC ^∞ closed manifold and Diff¹ (M) be the space of diffeomorphisms ofM endowed with theC ¹ topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC ¹ interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC ¹ interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.     

A deformation lemma on aC 1 manifold

The classical construction of deformations by mean of pseudo-gradient vector fields requires theC ^1,1 regularity. Here, we are concerned with a deformation lemma for aC ¹ function on a manifold defined by aC ¹ functional. We will assume some coupled Palais-Smale conditions between the two functions. The deformation is constructed with the help of integral lines of pseudo-gradient vector fields on a foliation of the manifold. Three different constructions are used for a sub-manifold of codimension 1 in finite dimension, then in infinite dimension and lastly a sub-manifold of any finite codimension in an infinite dimensional Banach space.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

Applications of the group analysis of differential equations to some systems of noncommuting <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth vector fields

Given a canonical basis of C ¹-smooth vector fields \(\{ \tilde X_i \} \) satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to \(\{ \tilde X_i \} \).     

Proof of Gromov’s theorem on homogeneous nilpotent approximation for vector fields of class C 1

The article is devoted to the asymptotic properties of the vector fields $\tilde X_i^g $ , i = 1, …, N, θ _g -connected with C ¹-smooth basis vector fields {X _i } _i=1,…,N satisfying condition (+ deg). We prove a theorem of Gromov on the homogeneous nilpotent approximation for vector fields of classC ¹. Nontrivial examples are constructed of quasimetrics induced by vector fields {X _i } _{i=1, …, N} .     

On Ω-blow-ups in the simplest <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth skew products of interval mappings

The authors prove a criterion (necessary and sufficient condition) for the emergence of the C ⁰-Ω-blow-up for C ¹-smooth skew products of interval mappings with closed set of periodic points. An example of the mapping with given properties that admits the C ⁰-Ω-blow-up is presented. It is proved that the C ¹-Ω-blow-up is impossible for mappings of such a type (in the space of C ¹-smooth skew products of interval mappings). It is proved that there is no one-parameter family of C ¹-smooth skew products of interval mappings with closed set of periodic points C ¹-smoothly depending on the parameter in which from one fixed point, periodic orbits with periods 2 and 4 simultaneously arise. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Hölder continuity for a class of strongly degenerate Schrödinger operator formed by vector fields

In this paper we obtain the Hölder continuity property of the solutions for a class of degenerate Schrödinger equation generated by the vector fields: $$- \sum\limits_{i,j = 1}^m {X_j^* \left( {a_{ij} \left( x \right)X_i u} \right) + \vec bXu + vu = 0,}$$ where X = {X ₁, ...,X _m } is a family of C ^∞ vector fields satisfying the Hörmander condition, and the lower order terms belong to an appropriate Morrey type space.     

Smooth extension of functions on a certain class of non-separable Banach spaces

Let us consider a Banach space X with the property that every real-valued Lipschitz function f can be uniformly approximated by a Lipschitz, C¹-smooth function g with Lip(g)?CLip(f) (with C depending only on the space X). This is the case for a Banach space X bi-Lipschitz homeomorphic to a subset of c₀(Γ), for some set Γ, such that the coordinate functions of the homeomorphism are C¹-smooth (Hájek and Johanis, 2010 [10]). Then, we prove that for every closed subspace Y⊂X and every C¹-smooth (Lipschitz) function f:Y→R, there is a C¹-smooth (Lipschitz, respectively) extension of f to X. We also study C¹-smooth extensions of real-valued functions defined on closed subsets of X. These results extend those given in Azagra et al. (2010) [4] to the class of non-separable Banach spaces satisfying the above property.