Title -
Perturbing Misiurewicz parameters in the exponential family
Abstract:
In one-dimensional real and complex dynamics, a map whose
post-singular (or post-critical) set is bounded and uniformly
repelling is often called a Misiurewicz map. In results hitherto,
perturbing a Misiurewicz map is likely to give a ('chaotic')
non-hyperbolic map, as per Jakobson's Theorem for unimodal interval
maps. This is despite the hyperbolic parameters forming an open, dense
set (at least in the interval setting). We shall present some
background results and explain why the contrary holds in the complex
exponential family $z \mapsto \lambda \exp(z)$: Misiurewicz maps are
Lebesgue density points for hyperbolic parameters.