Toshiaki Adachi

Holomorphic helices in a complex space form

In a complex space form M we shall investigate a smooth curve γ which is generated by a holomorphic Killing vector field X on M .

Twisted Perron-Frobenius theorem and L-functions

Abstract A theorem of Perron-Frobenius type and its twisted version are established in a setting of topological graphs. The applications include a partial extension of Selbergs results on his zeta functions and a result by Parry and Pollicott on meromorphic continuations of dynamical zeta functions to certain L -functions associated to a dynamical system of Anosov type.

Lamination of the moduli space of circles and their length spectrum for a non-flat complex space form

A smooth curveγ : R → parametrized by its arclength on a complete Riemannian manifold is called acircle of geodesic curvatureκ if it satisfies the differential equation∇γ̇∇γ̇ γ̇( ) = −κ2γ̇( ) Hereκ is a non-negative constant and ∇γ̇ denotes the covariant differentiation along γ with respect to the Riemannian connection on . When κ = 0, asγ is parametrized by its arclength, this equation is equivale nt to the equation of geodesics. In this paper we study the set of congr ue ce classes of circles on a non-flat complex space form, which is either a complex pro jective spaceC or a complex hyperbolic space C . We call two circlesγ1 and γ2 on arecongruent if there exist an isometryφ of and a constant0 satisfyingγ1( ) = φ ◦ γ2( + 0) for all . We denote by Cir( ) the set of all congruence classes of ci rcles on . In the preceding papers [5] and [3], we studied length spectu m of circles on nonflat complex space forms. We call a circle γ closed if it satisfies γ( ) = γ( + ) for every with some positive constant . The minimum positive wit h this property is called thelength of γ and is denoted by length( γ). For an open circleγ, a circle which is not closed, we set length( γ) = ∞. The length spectrumL : Cir( ) → R ∪ {∞} of circles is defined byL([γ]) = length(γ), where [γ] denotes the congruence class containingγ. In these papers [5], [3], we find that the moduli spaces Cir( C ) and Cir(C ) of circles on non-flat complex space forms have a natural lam in tion structure: If we restrict the length spectrum L on each leaf, it is continuous. In the first half of this paper we study the phenomenon of circles at t he boundary of each leaf. For a sequence {σι} of closed curvesσι : 1 = [0 1]/∼ → on we shall call limσι its limit curve if it exists. We study this lamination from th e viewpoint of limit curves of circles. The second half of this paper is devoted to add some resluts on length functions of circles on non-flat complex space forms. As two circles hav e the same geodesic

Markov families for Anosov flows with an involutive action

The aim of this note is to construct “involutive” Markov families for geodesic flows of negative curvature. Roughly speaking, a Markov family for a flow is a finite family of local cross-sections to the flow with fine boundary conditions. They are basic tools in the study of dynamical systems. In 1973, R. Bowen [5] constructed Markov families for Axiom A flows. Using these families, he reduced the problem of counting periodic orbits of an Axiom A flow to the case of hyperbolic symbolic flows.

Distribution of closed geodesics with a preassigned homology class in a negatively curved manifold

Let M be a compact Riemannian manifold whose geodesic flow φ i : UM→UM on the unit tangent bundle is of Anosov type. In this paper we count the number of φ i -closed orbits and study the distribution of prime closed geodesies in a given homology class in H 1 ( M, Z ). Here a prime closed geodesic means an (oriented) image of a φ i -closed orbit by the projection p : UM → M .

The Euclidean factor of a Hadamard manifold

The ideal boundary -Y(oo) of a Hadamard manifold X is the set of asymptotic classes of rays on X. We shall characterize the Euclidean factor of X by information on X(oo). Under the assumption that the diameter of .Y(oo) is n , we call a boundary point that has a unique point of Tits distance n a polar point. We shall show that such points form a standard sphere and compose the boundary of the Euclidean factor of the given Hadamard manifold.

EXTRINSIC CIRCULAR TRAJECTORIES ON GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

We say a trajectory for a Sasakian magnetic field on a geodesic sphere in a complex projective space to be extrinsic circular if it can be seen as a circle in the ambient space. We study how the moduli space of extrinsic circular trajectories behaves in the moduli space of all circles in the ambient complex projective space. As an application we characterize the geodesic sphere of special radius which lies on the boundary position of the family of Berger spheres among all geodesic spheres and that has a characteristic properties from the viewpoint of lengths of circles.

CHARACTERISATIONS OF GEODESIC HYPERSPHERES IN A NON-FLAT COMPLEX SPACE FORM

Totally η-umbilic real hypersurfaces are the simplest examples of real hypersurfaces in a non-flat complex space form. Geodesic hyperspheres in this ambient space are typical examples of such real hypersurfaces. We characterise every geodesic hypersphere by observing the extrinsic shapes of its geodesics and using the derivative of its contact form.