Attractor and Repeller Points for a Several-Variable Analytic Dynamical System in a Non-Archimedean Setting

We study discrete several-variable analytic dynamical systems over a complete non-Archimedean field with a nontrivial valuation and give sufficient conditions for a fixed point of the system to be an attractor, a repeller, or an indifferent point.     

Function solutions to certain Diophantine equations

We investigate whether certain Diophantine equations have or have not solutions in entire or meromorphic functions defined on a non-Archimedean algebraically closed field of characteristic zero. We prove that there are no non-constant meromorphic functions solving the Erdös–Selfridge equation except when the corresponding curve is a conic. We also show that there are infinitely many non-constant entire solutions to the Markoff–Hurwitz equation.     

The stability of additive $(\alpha,\beta)$-functional equations

In this paper, we investigate the following $(\alpha,\beta)$-functional equations $$ 2f(x)+2f(z)=f(x-y)+\alpha^{-1}f(\alpha (x+z))+\beta^{-1}f(\beta(y+z)),~~~~~~~~~(0.1) $$ $$ 2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z)) +\beta^{-1}f(\beta(y-z)),~~~~~~~~~~~(0.2) $$ where $\alpha,\beta$ are fixed nonzero real numbers with $\alpha^{-1}+\beta^{-1}\neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the $(\alpha,\beta)$-functional equations $(0.1)$ and $(0.2)$ in non-Archimedean Banach spaces.     

Linear Forms in two m-adic Logarithms and Applications to Diophantine Problems

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (x ⁿ – 1)/(x – 1) = y ^q .     

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series (t,x) satisfying the conditions (0,x)=x and (F(s,t),x)=(s,(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows with any specified divisor of fixed points, where a point of the open unit disk is a fixed point of order n if (x–) ⁿ |((t,x)–x). Furthermore, if is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow , an element of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of (x) is equal to (,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.     

概率收缩偶与非阿基米德Menger概率赋范空间中非线性方程组的解*

本文在非阿基米德Menger概率赋范空间中引入了概率收缩偶的概念,研究了非阿基米德Menger概率赋范空间中具概率收缩偶的非线性方程组的解的存在性与唯一性.发展和改进了引文[1～5]的相应结果.     

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L₂-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).     

Functional inequalities in non-Archimedean normed spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in non-Archimedean Banach spaces.     

Fourier series, measures and divided power series in the theory of function fields

Much of classical number theory is based on Fourier series. Such series play a vital role in the study of characteristic-0 zeta-functions: In the complex theory one has theta-series and Tate's thesis. In the p-adic theory one has Mahler's theorem on binomial coefficients which is used to déscribe the ring of p-adic measures. In this paper, we discuss a version of binomial coefficients for function fields due to L. Carlitz. We will show how these functions arise naturally out of gamma functions for function fields. We will also use some work of C. Wagner to establish that the ring of -adic measures is canonically isomorphic to the ring of divided power-series. The computation of these power-series in specific instances is now an important problem in the theory. Finally, we show the existence of many Fourier transforms in the -adic theory. The explicit computation of these would also be very interesting.Partially supported by NSF grant DMS-8521678. Current address: Department of Mathematics, UMBC, MD 21228, U.S.A.Dedicated to L. Carlitz