Toshikazu Sunada

Closed orbits in homology classes

© Publications mathématiques de l’I.H.É.S., 1990, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Standard realizations of crystal lattices via harmonic maps

An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.

On the spectrum of periodic elliptic operators

It was observed in [Su5] that the spectrum of a periodic Schrodinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators. Let X be a Riemannian manifold of dimension n on which a discrete group F acts isometrically, effectively, and properly discontinuously. We assume that the quotient space F \ X (which may have singularities) is compact. Let E be a F-equivariant hermitian vector bundle over X, and D : C°°(E) • C°°(E) a formally self-adjoint elliptic operator which commutes with the F-action. For short, we call such a D a F-periodic operator. It is easy to show (see Section 1) that the symmetric operator D with the domain C^{E) is essentially self-adjoint, so that D has a unique self-adjoint extension in the Hilbert space L(E) of square integrable section of E, which we denote also by D by a slight abuse of notation. Let Cfed(F, # ) denote the tensor product of the reduced group C*-algebra of F with the algebra # of compact operators on a separable Hilbert space of infinite dimension, and by Xir the canonical trace on C*ed(F, # ) . We define the Kadison constant C (F) by C(F) =inf {trrP ; P is a non-zero projection in C*ed(F, # ) } . By definition, F is said to satisfy the Kadison property if C(F) > 0. It is a conjecture proposed by Kadison that, if F is torsion free, then C(F) — 1. A

Asymptotic behavior of quantum walks on the line

Abstract This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the ‘normalized’ position of the walk. When the position is in the support of the weak-limit distribution obtained by Konno (2005) [5] , one observes, in addition to the limit distribution itself, an oscillating phenomenon in the leading term of the asymptotic formula. When the position lies outside of the support, one can establish an asymptotic formula of large deviation type. The rate function, which expresses the exponential decay rate, is explicitly given. Around the boundary of the support of the limit distribution (called the ‘wall’), the asymptotic formula is described in terms of the Airy function.

Spherical means and geodesic chains on a Riemannian manifold

Some spectral properties of spherical mean operators defined on a Riemannian manifold are given. As an application we deduce a statistic property of geodesic chains which is interesting from the view point of geometric probability.

Discrete Schrödinger operators on a graph

In this paper, we study some spectral properties of the discrete Schrodinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph. The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

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.

Spectrum of a compact flat manifold

Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen fur nicht-kommerzielle Zwecke in Lehre, Forschung und fur die private Nutzung frei zur Verfugung. Einzelne Dateien oder Ausdrucke aus diesem Angebot konnen zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung moglich. Die Rechte fur diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.

Holomorphic mappings into a compact quotient of symmetric bounded domain

In this paper, we shall be concerned with the finiteness property of certain holomorphic mappings into a compact quotient of symmetric bounded domain. Let be a symmetric bounded domain in n -dimensional complex Euclidean space C n and Γ\ be a compact quotient of S by a torsion free discrete subgroup Γ of automorphism group of . Further, we denote by l ( ) the maximum value of dimension of proper boundary component of , which is less than n (=dim ).

Homology of closed geodesics in certain Riemannian manifolds

It is shown, by using the trace formula of Selberg type, that every primitive, one-dimensional homology class of a negatively curved compact locally symmetric space contains infinitely many prime closed geodesics. 0. Let M be a compact space form of a symmetric space of rank one. In this note, we prove that each homology class in H1 (M, Z) contains infinitely many free homotopy classes of closed curves, that is, the mapping induced from the Hurewicz homomorphism [XI (M)] -? H1 (M, Z) is an oo-to-one correspondence. One of the geometric consequences is that any primitive homology class contains infinitely many prime closed geodesics, since, as was shown by Hadamard, every nonnull homotopy class contains a closed geodesic which is automatically prime if the homology class is primitive. Here a homology class a is called primitive if Ol is not a nontrivial integral multiple of another homology class. If dimM = 2, then one can prove the much stronger assertion that every homology class contains infinitely many prime closed geodesics (see ?2). The following theorem, which can be shown by means of a number-theoretic argument applied to the L-functions associated to length spectrum of closed geodesics (see [1, 4] for proof), is somewhat related to the result. THEOREM. Let H be a subgroup of H1(M, Z) of finite index, and let al be a coset in Hi/H. If M is negatively curved, then there exist infinitely many prime closed geodesics whose homology classes are in ca. 1. The proof relies heavily on the trace formula for the heat kernel function. We shall start with a general setting. Let 7r: M -? M be the universal covering of a compact Riemannian manifold M. The fundamental group ri (M) acts on M in the usual way. For brevity we write r for 7ri(M). For an element -y in r, we denote by rF the centralizer of -y, and by [-y] the conjugacy class of -y. The set of all conjugacy classes is denoted by [r]. Let p: r -, U(N) be a unitary representation, and let Ep be the flat vector bundle associated to p. We denote by AP the Laplacian acting on the sections of Ep. The fundamental solution of the heat equation on M will be denoted by k(t; x, y). The following lemma is proved in the same way as the proof of the Selberg trace formula (see [6]). LEMMA 1. tr(e-tAP) E tr p(y)J k(t; x, ay) dx. [I]E[r] Mr Received by the editors January 22, 1985. 1980 Mathematics Subject Classification. Primary 53C35, 53C22. tSupported by the Ishida Foundation. ?1986 American Mathematical Society 0002-9939/86