 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 $f$ with mathematical notation if there exist distinct n and m such that $f^{n}(x)=f^{m}(x)$(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 $1$. 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 $z^2-z$.”