P-adic Systems Research Articles

Qualitative theory of p-adic dynamical systems

We consider a class of dynamical systems over the p-adic number field: hierarchical dynamical systems. We prove a strong variant of the Poincare theorem on the number of returns for such systems and show that hierarchical systems do not admit mixing. We describe hierarchical dynamical systems over the projective line and present an example of a nonhierarchical p-adic system that admits mixing: the p-adic baker’s transformation.

Models of p-adic mechanics

We segregate the class of ultrametric (p-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincare recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum p-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.

Adelic integrable systems

Incorporating the zonal spherical function (zsf) problems on real and p-adic hyperbolic planes into a Zakharov–Shabat integrable system setting, we find a wide class of integrable evolutions that respect the number-theoretic properties of the zsf problem. This means that at all times these real and p-adic systems can be unified into an adelic system with an S matrix that involves (Dirichlet, Langlands, Shimura,...) L functions.

P-adic number theory

p-adic systems, although introduced by Hensel in 1908, have only recently attracted attention for their possible uses in exact linear computations, matrix processors, signal transformations and cryptography. The paper gives a detailed analysis of the subject of p-adic number theory. An examination of both the infinite and finite p-adic number systems is presented. For infinite systems, a practical efficient algorithm is developed for the computation of the p-adic period for any rational number, given the prime p. In finite systems, on the other hand, we demonstrate that the original algorithms developed by Krishnamurthy for the four main arithmetic operations are erroneous and we present new algorithms which circumvent these drawbacks.