Rational preperiodic points of rational maps

In this thesis we focus on the preperiodic and periodic points of the dynamicalsystems associated to rational maps of degree 2. We discuss the state of the artregarding rational preperiodic points of polynomials of degree 2 defined over therational field with an emphasis on the Uniform Boundedness Conjecture on thenumber of such points. We then study the known classification results of rationalpreperiodic points of rational maps of degree 2, these maps include rational mapsof degree 2 with abelian automorphism groups and rational maps of degree 2 witha rational periodic critical point of period 2.