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, $latex p$-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 $latex f$ on a set $latex X$; $latex f:X\to X$ a point x in X is called preperiodic point if it has finite forward orbit under iteration of $latex f$ with mathematical notation if there exist distinct n and m such that $latex f^{n}(x)=f^{m}(x)$(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 $latex 1$. It is a more precise version of I.N. Baker's theorem which statesLet $ $latex P$ be a polynomial of degree at least two and suppose that $latex P$ has no periodic points of period $latex n$. Then $latex n=2$ and $latex P$ is conjugate to $latex z^2-z$.''