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, -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 on a set ; a point x in X is called preperiodic point if it has
finite forward orbit under iteration of with mathematical notation if there exist distinct n
such that (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 . It is a more precise version of I.N. Baker's theorem which statesLet $ be a polynomial of degree at least two and suppose that has no periodic points of period . Then and is conjugate to .”