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**11 - 15**of**15**### What are p-Adic Numbers? What are They Used for?

, 2013

"... In this short paper we give a popular introduction to the theory of p-adic numbers. We give some properties of p-adic numbers distinguishing them to “good” and “bad”. Some remarks about applications of p-adic numbers to mathematics, biology and physics are given. ..."

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In this short paper we give a popular introduction to the theory of p-adic numbers. We give some properties of p-adic numbers distinguishing them to “good” and “bad”. Some remarks about applications of p-adic numbers to mathematics, biology and physics are given.

### BOOK REVIEW

"... Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on ..."

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Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on a manifold, and the local properties of such a map are largely governed by the behaviour of the derivative. Thus one quickly arrives at the simplest model system to study: iterates of a linear map on R n. Specializations and other model systems abound; those most relevant here are rational maps on the Riemann sphere (complex dynamics), attractors for expansive maps, automorphisms of compact groups (algebraic dynamics) and iteration of polynomials in an arithmetic setting. Each of these contributes motivation, ideas, and suggestions for analogies, to the topics in this book. 1. p-adic dynamics in vivo Before turning to the willful pursuit of p-adic dynamics, it is important to recognize that p-adic phenomena cannot be avoided – they arise in the familiar setting of smooth maps on manifolds. The map

### BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

"... Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on ..."

Abstract
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(Show Context)
Dynamical systems most naturally arise in the setting of trajectories (time orbits) constrained to lie in a phase space, typically a manifold, of a system evolving according to some physical rules. Sampling such a flow at fixed time intervals often reduces the problem to the study of a smooth map on a manifold, and the local properties of such a map are largely governed by the behaviour of the derivative. Thus one quickly arrives at the simplest model system to study: iterates of a linear map on R n. Specializations and other model systems abound; those most relevant here are rational maps on the Riemann sphere (complex dynamics), attractors for expansive maps, automorphisms of compact groups (algebraic dynamics) and iteration of polynomials in an arithmetic setting. Each of these contributes motivation, ideas, and suggestions for analogies to the topics in this book. 1. p-adic dynamics in vivo Before turning to the willful pursuit of p-adic dynamics, it is important to recognize that p-adic phenomena cannot be avoided – they arise in the familiar setting of smooth maps on manifolds. The map