Abstract:

In this talk I will first make a historical outline on the techniques and results of perturbing maps by parameter exclusion, originally founded by M. Benedicks and L. Carleson in the 1980s. The starting map usually has strong expanding properties, e.g. its critical set is non-recurrent or the map is even postcritially finite, such that all critical points end up in repelling cycles (a so called Misiurewicz-Thurston map). They are special cases of non-uniformly hyperbolic maps, so called Collet-Eckmann maps on the Riemann sphere, which are maps $f$ for which the derivative of the n:th iterate grows exponentially along the forward orbit of the critical values $f(c)$, where $c$ is a critical point on in the Julia set (assuming that no critical point lies in the forward orbit of $c$). Extending the old traditional results by Benedicks and Carleson and combining it with, among other things, more recent results by G. Levin, I will state some new results on perturbing more general Collet-Eckmann maps where we allow the critical set to be (slowly) recurrent.