Topological invariance of generic non-uniformly expanding multimodal maps

We introduce a \emph{Topological Slow Recurrence} condition on the orbits of critical points of multimodal maps and show that this condition is equivalent to the simultaneous occurrence of metric \emph{Slow Recurrence} and \emph{Collet-Eckmann} conditions. In particular, if \( f \) is non-uniformly expanding and the critical points are generic with respect to the absolutely continuous invariant measures, then any map \( g \) topologically conjugate to \( f \) is also non-uniformly expanding.

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Published 1 January 2006