  This paper is devoted to study the topological invariance of several
non-uniform hyperbolicity conditions of one-dimensional maps. In contrast with
the case of maps with only one critical point, it is known that for maps with
several critical points the Collet-Eckmann condition is not in itself
invariance under topological conjugacy. We show that the Collet-Eckmann
condition together with any of several slow recurrence conditions is invariant
under topological conjugacy. This extends and gives a new proof of a result by
Luzzatto and Wang that also applies to the complex setting.
