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Arithmetical Dynamics: An intro:

Arithmetic dynamics is a field that amalgamates two areas of mathematics, discrete dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.

Given an endomorphism f on a set X; f:X\to X a point x in X is called preperiodic point if it has finite forward orbit under iteration of   <code>f with mathematical notation if there exist distinct n and m such that <code>f^{n}(x)=f^{m}(x)(i.e it is eventually periodic''). We are trying to classify for which maps are there not as many preperiodic points as their should be in dimension   <code>1. It is a more precise version of I.N. Baker's theorem which statesLet $ P be a polynomial of degree at least two and suppose that P has no periodic points of period n. Then n=2 and P is conjugate to z^2-z.''

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