wild recurrent critical points

Abstract

it is conjectured that a rational map whose coefficients are algebraic over $\mathbb q_p$ has no wandering components of the fatou set. benedetto has shown that any counterexample to this conjecture must have a wild recurrent critical point. we provide the first examples of rational maps whose coefficients are algebraic over $\mathbb q_p$ and that have a (wild) recurrent critical point. in fact, it is shown that there is such a rational map in every one-parameter family of rational maps that is defined over a finite extension of $\mathbb q_p$ and that has a misiurewicz bifurcation.