Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Let phi is an element of 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 phi is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height (h) over cap (phi)(x) of a nonpreperiodic rational point x is bounded below by a uniform multiple of the height of phi 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.