Publications (21)14.17 Total impact
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ABSTRACT: We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact threemanifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an areapreserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.  [Show abstract] [Hide abstract]
ABSTRACT: We consider semigroup actions on the unit interval generated by strictly increasing $C^r$maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the nonlinearity and the Schwarzian derivative, the above semigroups exhibit, $C^r$generically for any r, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a $C^r$generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.  [Show abstract] [Hide abstract]
ABSTRACT: We construct a nonergodic maximal entropy measure of a C ∞ diffeomorphism with a positive entropy such that neither the entropy nor the large deviation rate of the measure is influenced by that of ergodic measures near it.  [Show abstract] [Hide abstract]
ABSTRACT: We study the growth of the numbers of critical points in onedimensional lattice systems by using (real) algebraic geometry and the theory of homoclinic tangency. 
Article: Homotopy classes of total foliations
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ABSTRACT: On every compact and orientable threemanifold we construct total foliations (three codimensionone 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 bicontact structures.  [Show abstract] [Hide abstract]
ABSTRACT: An analog of the BaumslagSolitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure. 
Article: Remark on dynamical Morse inequality
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ABSTRACT: We solve a transversality problem relating to BertelsonGromov’s “dynamical Morse inequality”.  [Show abstract] [Hide abstract]
ABSTRACT: We show the local rigidity of the natural action of the Borel subgroup of SO_+(n,1) on a cocompact quotient of SO_+(n,1) for n>2.  [Show abstract] [Hide abstract]
ABSTRACT: This is a note of the author's lectures at "Advanced courses in Foliation" in the research program "Foliation", which was held at the Centre de Recerca Mathematica in the May of 2010. In this note, we discuss about the relationship between deformation of actions of Lie groups and the leafwise cohomology of the orbit foliation. 
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ABSTRACT: For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits C 1 persistent homoclinic 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.  [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.  [Show abstract] [Hide abstract]
ABSTRACT: On every compact and orientable threemanifold, 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 bicontact 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 threedimensional 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 any topologically transitive codimensionone 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 codimensionone Anosov flow is topologically transitive. Recently, Simic showed that any higher dimensional codimensionone 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 codimensionone 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 threedimensional manifolds up to smooth conjugacy. As a corollary, there exists a nonhomogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.  [Show abstract] [Hide abstract]
ABSTRACT: We show that if a C 2 codimensionone foliation on threedimensional 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 threedimensional 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 2regular 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 × Imodels.  [Show abstract] [Hide abstract]
ABSTRACT: We give complete classification of C^2regular and nondegenerate 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.
Publication Stats
57  Citations  
14.17  Total Impact Points  
Top Journals
Institutions

20032013

Kyoto University
 Department of Mathematics
Kioto, Kyōto, Japan


2002

The University of Tokushima
Tokusima, Tokushima, Japan
