Let M be a compact manifold without boundary. Let f: M → M be a C1 diffeomorphism. Then the nonwandering set Ω(f) is defined to be the closed invariant set consisting of x ∈ M such that for any neighborhood U of x, there exists an integer n ≠ 0 satisfying fn(U) ∩ U ≠ ø. In particular, the set Per (f) of all periodic points is included in Ω(f).

The purpose of the present paper is to give a complete proof of the theorem which will be used in a paper of the second author [4].

We will discuss certain spectral properties of selfadjoint ordinary differential operators of the form iA(d/dx) + V acting in L2(R)n = Σ ⊕ L2(R)n, where A is a real diagonal constant matrix and V an Hermitian matrix valued function on R which satisfies some conditions to be stated in the sequel.

Let f be a Cr-diffeomorphism, r ≦ 1, on a compact differentiable manifold M with dim M ≧ 2. In [9] F. Takens introduced the concept of extended f-orbits and conjectured the following.

Au 19e siècle le terme de géométrie algébrique désignait exclusivement la géométrie birationnelle. Aujourd’hui on n’accepterait pas cette définition, parce que l’objet de la géométrie algébrique est celui des invariants biréguliers plutôt que birationnels. Quelle que soit la définition adoptée, il est intéressant de connaître la structure de groupes des automorphismes birationnels. Mais celle-ci est très peu connue (Mumford [10]). La question suivante a été posée depuis longtemps; déterminer à conjugaison près tous les sous-groupes algébriques connexes et maximaux du groupe de Cremona Crn. Le groupe Crn est, par définition, le groupe des automorphismes du corps des fonctions rationnelles à n indéterminées. Si n = 1. Cr1 coïncide avec le groupe des automorphismes de P1 Enriques a résolu ce problème pour n = 2 (Enriques [5]). On croit généralement (par exemple Godeaux [8]) que Enriques et Fano l’ont fait pour n = 3 (Enriques et Fano [6]). Après avoir collaboré avec Enriques, G. Fano a travaillé seul sur ce problème et il a laissé des résultats très intéressants dont la plupart des démonstrations ne semble pas rigoureuse. Demazure [4] a étudie les; sous-groupes algébriques de rang maximum n de Crn. Il a démontré qu’il y a une correspondence bijective entre les sous-groupes algébriques de rang maximum n de Crn et les systèmes d’Enriques. Puisque les derniers sont de nature combinatoire, c’est un dictionaire géométrique-combinatoire.

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.

It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.

In a previous paper [6] we had some criteria for the prime decomposition in certain non-abelian extensions over the rational number field Q, and as its special case we had a reciprocity of the biquadratic residue symbol. The reciprocity was obtained by using a descent method of the prime decomposition for a central extension over Q which is abelian over a biquadratic field In the present paper we study on the case over a biquadratic field in general. We define a symbol [d1, d2, p] which expresses the decomposition law of a rational prime p in a central extension mentioned above.

In this paper we deal with the problem (C) in § 4 of [4]. Let Ik be the Shimura mapping in [4] of Sk(4N, χ) into k-1(N′ χ2) (see p. 458). The problem (C) can be stated as follows: I3(f) is a cusp form if and only if ‹f, h› = 0 for all h ∈ U, where U is the vector space spanned by every theta series of S3(4N, χ) associated with some Dirichlet character.

G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:

Theorem. It is necessary and sufficient for the class number of an imaginary quadratic number fieldD = 1 — 4m, m > 0, to be one that x2 — x + m is prime for any integer x such that 1 ≤ x ≤ m — 2.

Let K be an infinite field of characteristic not 2. Let qx, qY be nonsingular quadratic forms on vector spaces X, Y over K, respectively. Assume that there is a bilinear map B:X × Y → Y such that qY(B(x, y)) = qx(x)qY(y). To each such triple {qx, qY, B} one associates the Hopf map h: Z = X × Y → W = K × Y by h(z) = (qx(x) – qY(y), 2B(x, y)), z = (x, y).

Let k be an algebraic number field of finite degree over the field Q of rational numbers. We denote by o the ring of integers in k. In general, for a subring A, containing 1, of a universal domain Ω we denote by GL(n, A) the subgroup of GL(n, Ω) consisting of matrices x = (xij) with xij ∈ A and det x ∈ A×, the group of units of A. Now, we consider an algebraic group G in GL(n, Ω) defined over k.

In this paper, we study topologically stable homeomorphisms of the circle. Our results are the following:

Theorem 1. A homeomorphism of the circle is topologically stable if and only if it is topologically conjugate to some Morse-Smale diffeomorphism.

Theorem 2. There exists a homeomorphism of the circle which has the pseudo-orbit-tracing-property but is not topologically stable.

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.

1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;

(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.