Small dynamical heights for quadratic polynomials and rational functions

Let $f\in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman’s Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height ${\widehat{h}}_f(x)$ of a non-preperiodic rational point $x$ is bounded below by a uniform multiple of the height of $f$ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range of previous searches.