Perturbing Misiurewicz Parameters in the Exponential Family

Abstract

In one-dimensional real and complex dynamics, a map whose postsingular (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 non-hyperbolic map, as per Jakobson's Theorem for unimodal interval maps. This is despite genericity of hyperbolic parameters (at least in the interval setting). We show the contrary holds in the complex exponential family z → λ exp(z): Misiurewicz maps are Lebesgue density points for hyperbolic parameters. As a by-product, we also show that Lyapunov exponents almost never exist for exponential Misiurewicz maps. The lower Lyapunov exponent is −∞ almost everywhere. The upper Lyapunov exponent is non-negative and depends on the choice of metric.

Semi-Hyperbolic Maps Are Rare

“…equal to the dimension of the parameters space. For the exponential family f λ (z) = λe z , λ ∈ C similar results also hold, see [3] and [5]. We prove in this paper the following.…”