R. Bowen and P. Walters [2] have defined expansive flows on metric spaces which generalized the similar notion by D. Anosov [1]. On the other hand, P. Walters [4] investigated continuous transformations of metric spaces with discrete centralizers and unstable centralizers and proved that expansive homeomorphisms have unstable centralizers.

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

It seems to the author that the theory of vector bundles of rank r over an abelian variety of dimension g is not sufficiently developed except for the two cases r = 1 and g = 1. The case r = 1 is the theory of theta functions and is one of the richest branches of mathematics. It is quite natural to try to explore vector bundles in the higher rank case as we did in the case r = 1.

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

Let f(z) be a cusp form of type (l,ε) on Γ0(N) which is a common eigenfunction of all Hecke operators. For such f(z), Deligne and Serre [1] proved that there exists a linear representation

such that the Artin L-function for p is equal to the L-function associated to f(z).

Let be the space of cusp forms of weight k with respect to SL(2, Z). Let p be a prime number and let Tk(p) be the Hecke operator of degree p acting on as a linear endomorphism. Put Hk(X) = det (I – Tk(p)X + pk-lX2I), where I is the identity operator on . Hk(X) is a polynomial with coefficients of rational integers, which is called the Hecke polynomial.

It is well known that for the family F of Riemann surfaces {R(z)} defined by the equations y2 = x(x — l)(x — z), zεC — {0,1}, we have one independent abelian differential ω = y−1dx on each R(z) and if we consider z as a parameter on C — {0,1}, the integrals are solutions of the Gauss’s differential equation

In the previous paper [3], the author generalized the uniqueness theorems of meromorphic functions given by G. Pólya in [5] and R. Nevanlinna in [4] to the case of meromorphic maps of Cn into the N- dimensional complex projective space PN(C).

In [15] Titchmarsh posed and solved under the generalized Riemann Hypothesis, the problem of an asymptotic behavior of the number of the solutions of the equation 1 = p — n1n2 for a prime p ≦ x and natural numbers n1 and n2. When we put then the above problem is to get an asymptotic law for the sum

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 Cn 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).