[Show abstract][Hide abstract] ABSTRACT: We study the growth of the numbers of critical points in one-dimensional
lattice systems by using (real) algebraic geometry and the theory of homoclinic
[Show abstract][Hide abstract] ABSTRACT: A triple ξ 1 ,ξ 2 ,ξ 3 of smooth transversely oriented plane fields on an (oriented, closed) 3-manifold M is a total plane field if ξ 1 (p)∩ξ 2 (p)∩ξ 3 (p)=0 at any p∈M. A total plane field is a foliation if each ξ i is integrable. Vanishing of Euler characteristics and second Stiefel-Whitney classes implies the existence of total plane fields on any closed, oriented 3-manifold. D. Hardorp [“All compact orientable three dimensional manifolds admit total foliations”, Mem. Am. Math. Soc. 233, 74 p. (1980; Zbl 0435.57005)] proved that each closed, orientable 3-manifold carries a total foliation. Moreover J. W. Wood [“Foliations on 3-manifolds”, Ann. Math. (2) 89, 336–358 (1969; Zbl 0176.21402)] proved that each plane field on a closed 3-manifold is homotopic to a foliation. In the paper under review it is proven that each total plane field on a closed, oriented 3-manifold is homotopic to a total foliation. (Using the Eliashberg-Thurston theorem this then also implies that any oriented plane field with Euler class zero is homotopic to positive and negative contact structures which form a bicontact structure.) The basic idea of this long and involved paper is that homotopy classes of total plane fields are determined by (differenences of) spin structures and Hopf degrees. Namely there are natural bijections between the set of total plane fields, the set of orthonormal frames on TM, and the set of continuous maps from M to SO(3). For two total plane fields ξ 0 i and ξ i let Φξ 0 i ,ξ i :M→SO(3) be the corresponding map. Because of π 1 SO(3)=ℤ/2ℤ this yields an element sξ 0 i ,ξ i ∈Homπ 1 M,ℤ/2ℤ=H 1 M,ℤ/2ℤ which is the difference of spin structures. If sξ 0 i ,ξ i =0, then Φξ 0 i ,ξ i admits a lift Φ ˜ξ 0 i ,ξ i :M→Spin3 whose mapping degree is Hξ 0 i ,ξ i , the difference of Hopf degrees. Two total plane fields ξ 0 i and ξ i are homotopic if and only if sξ 0 i ,ξ i =Hξ 0 i ,ξ i =0. Thus the proof of the main theorem is reduced to showing that one can find a total foliation for any spin structure and difference of Hopf degrees. The basic building block of the authors construction are so called ℛ-components. These are total foliations ℱ i of S 1 ×D 2 such that ℱ 3 is a thick Reeb component (i.e., with a trivially foliated neighborhood of the boundary torus) and such that ℱ 1 and ℱ 2 are generated by the kernel of dy-χydt and dx-χydt, respectively, where χ is such that 0<χx<1 if x∈1 2,3 2 and χx=0 otherwise. The authors give an alternative proof of Hardorp’s theorem which shows in particular that a total foliation ℱ i exists for each given spin structure and with the additional condition that the total foliation contains two unknotted ℛ-components R ± which are ±1-framed, respectively. The result implying the main theorem is then that for each n one can find a total foliation ℱ n i with Hℱ i ,ℱ n i =n (and actually such that the above additional condition holds). The authors first solve this problem (with the additional condition) for a special total foliation ℛ i on S 3 , namely they find ℛ n i for the total foliation ℛ i consisting of two -1-framed unknotted ℛ-components. The desired total foliation ℱ n on M is then obtained by gluing ℱ i and ℛ n i along the boundaries of the ℛ-components.
[Show abstract][Hide abstract] ABSTRACT: For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits C 1 -persistent homo-clinic tangency. It provides an open subset of the space of C 1 -diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.
Proceedings of the American Mathematical Society 01/2008; 136(2). · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We define invariants of two dimensional $C^2$ projectively Anosov diffeomorphisms. The invariants are defined by the topology of the space of circles tangent to an invariant subbundle and are preserved under homotopy of projectively Anosov diffeomorphisms. As an application, we show that the invariant subbundle is not uniquely integrable and two distinct periodic orbits exist if certain invariants do not vanish.
Journal of the Mathematical Society of Japan 07/2007; 59(3). · 0.64 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures. Comment: 27 pages, 13 figures. This is the final version. To appear in Comm. Math. Helv
[Show abstract][Hide abstract] ABSTRACT: We show that if a C2 codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S2×S1.
Topology and its Applications 04/2007; · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher dimensional codimension-one Anosov flow is topologically transitive. Recently, Simic showed that any higher dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flow up to topological equivalence in higher dimensions. In this second version, the order of the presentation of the proof is changed and some minor errors in the previous version is corrected. Comment: 27 pages, no figures
[Show abstract][Hide abstract] ABSTRACT: We classify smooth locally free actions of the real affine group on closed
orientable three-dimensional manifolds up to smooth conjugacy. As a corollary,
there exists a non-homogeneous action when the manifold is the unit tangent
bundle of a closed surface with a hyperbolic metric.
Annals of Mathematics 02/2007; · 2.82 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that if a C 2 codimension-one foliation on three-dimensional manifold admits a transversely contracting flow, then it must be the unstable foliation of an Anosov flow.
[Show abstract][Hide abstract] ABSTRACT: We give the complete classification of regular projectively Anosov flows on
closed three-dimensional manifolds. More precisely, we show that such a flow
must be either an Anosov flow or decomposed into a finite union of $T^2 \times
I$-models. We also apply our method to rigidity problems of some group actions.
[Show abstract][Hide abstract] ABSTRACT: We give a classification of C 2 -regular projectively Anosov flows on closed three dimensional manifolds. More precisely, we show that if the manifold is connected then such a flow must be either an Anosov flow or represented as a finite union of 𝕋 2 ×I-models.
Proceedings of the Japan Academy Series A Mathematical Sciences 12/2004; 80(10). · 0.39 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give complete classification of C^2-regular and non-degenerate
projectively Anosov flows on three dimensional manifolds. More precisely, we
prove that such a flow on a connected manifold must be either an Anosov flow or
represented as a finite union of $T^2 \times [0,1]$-models.
[Show abstract][Hide abstract] ABSTRACT: We define invariants of two dimensional projectively Anosov diffeomorphisms. More precisely, we show the space of circles tangent to the invariant subbundle has a kind of Morse decomposition and the homotopy type of its one point compactification is preserved under any homotopy of projectively Anosov diffeomorphisms. We also calculate the invariants for some examples.
Proceedings of the Japan Academy Series A Mathematical Sciences 10/2002; 78(8). · 0.39 Impact Factor