[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: 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). · 0.61 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 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 01/2007; · 0.51 Impact Factor
[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.
[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.