Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

We prove that a volume-preserving three-dimensional flow can be C¹-approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C¹-topology.     

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.     

Branched spines and Heegaard genus of 3-manifolds

We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S ³ and S ² × S ¹. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.     

Transitively twisted flows of 3-manifolds

A non-singular C ¹ vector field X of a closed 3-manifold M generating a flow induces a flow of the bundle N X orthogonal to X. This flow further induces a flow of the projectivized bundle of N X. In this paper, we assume that the projectivized bundle is a trivial bundle, and study the lift of to the infinite cyclic covering . We prove that the flow is not minimal, and construct an example of such that has a dense orbit. If is almost periodic and minimal, then is shown to be classified into three cases: (1) All the orbits of are bounded. (2) All the orbits of are proper. (3) is transitive. Received: December 2, 1999     

On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g ₀. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.     

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.     

Multivalued contractions

LetE be a Banach space,C a closed convex subset ofE, F a multivalued contraction fromC to itself with closed values. Ifx ₀ is a fixed point and ifF(x ₀) is not a singleton, then there exists a fixed pointx ₁ ofF which is different fromx ₀. We prove also that there is in the Euclidean space ² a multivalued contraction with compact connected values having a nonconnected set of fixed points.     

Sources and sinks for Axiom A flows on closed 3-manifolds

We prove that an Axiom A vector field on an orientable closed 3-manifold not homeomorphic toS³ for which every transverse torus bounds a solid torus either is transitive or has a sink or a source. This result is false without these hypotheses.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

On the number of singularities,zero curvature points and vertices of a simple convex space curve

We prove a generalization of the 4 vertex theorem forC ³ closed simple convex space curves including singular and zero curvature points.Work partially supported by CNPq. The second author is also grateful to the Universidade Federal de Viçosa (Brasil) for hospitality during the production of this work.     

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.     

Homogeneous continua in 2-manifolds

In 1960 R.H. Bing [2] proved that every homogeneous plane continuum that contains an arc is a simple closed curve. At that time Bing [2, p. 228] asked if every 1-dimensional homogeneous continuum that contains an arc and lies on a 2-manifold is a simple closed curve. We prove that no 2-manifold contains uncountably many disjoint triods. We use this theorem and decomposition theorems of F.B. Jones [10] and H.C. Wiser [19] to answer Bing's question in the affirmative. We also prove that every homogeneous indecomposable continuum in a 2-manifold can be embedded in the plane. It follows from this result and another theorem of Wiser [20] that every homogeneous continuum that is properly contained in an orientable 2-manifold is planar.     

Sur les feuilletages géodésiques continus des variétés hyperboliques

Resumé On démontre qu'une 3-variété hyperbolique compacte n'admet pas de feuilletageC ⁰, dont les feuilles sont géodésiques.

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Summary We prove that a compact hyperbolic 3-manifold does not possess aC ⁰ foliation, with geodesic leaves.

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Oblatum II-1993     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

Approximating <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-foliations by contact structures

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C ⁰-approximated by both positive and negative contact structures unless all leaves of the foliation are simply connected. As applications we deduce that the existence of a taut C ⁰-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C ⁰-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.     

On Birkhoff Theorems for Lagrangian invariant tori with closed orbits

We show that a C¹ torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T², and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T², shows that the second Birkhoff Theorem holds for C¹ Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T² and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection. Partially supported by CNPq, FAPERJ, TWAS     

Morse and semistable stratifications of K?hler spacesby \Bbb C*{\Bbb C}^{\ast}-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \Bbb C^*{\Bbb C}^{\ast} -K?hler manifold coincide if the moment map is proper and if the fixed points set X^{\Bbb C*}X^{{\Bbb C}^{\ast}} has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.     

On the dynamical coherence of structurally stable 3-diffeomorphisms

We prove that each structurally stable diffeomorphism f on a closed 3-manifold M ³ with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.