Perturbing Misiurewicz parameters in the exponential family

TL;DR

This paper demonstrates that in the complex exponential family, Misiurewicz maps are densely surrounded by hyperbolic maps, contrasting real dynamics, and reveals that Lyapunov exponents typically do not exist for these maps.

Contribution

It proves that Misiurewicz parameters are Lebesgue density points for hyperbolic parameters in the exponential family, a contrast to real dynamics, and analyzes Lyapunov exponents behavior.

Findings

Misiurewicz maps are Lebesgue density points for hyperbolic maps in the exponential family.

Lyapunov exponents almost never exist for exponential Misiurewicz maps.

Lower Lyapunov exponent is negative infinity almost everywhere.

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 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 $z \mapsto \lambda \exp(z)$: 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 negative infinity almost everywhere. The upper Lyapunov exponent is non-negative and depends on the choice of metric.