Perturbations of weakly expanding critical orbits

Genadi Levin

Published 2011-11-27, updated 2013-09-15Version 2

Let f be a polynomial or a rational function which has r summable critical points. We prove that there exists an r-dimensional manifold in an appropriate space containing f such that for every smooth curve in it through f, the ratio between parameter and dynamical derivatives along forward iterates of at least one these summable points tends to a non-zero number.