Local Critical Perturbations of Unimodal Maps

Abstract

We introduce a new complete metric in the space \({\mathcal {V}_2}\) of unimodal C 2-maps of the interval, with two maps close if they are close in the C 2-metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for \({\mathcal {V}_2}\) and show that structurally stable maps form an open dense subset of \({\mathcal {V}_2}\).