The Lyapunov Ground State of the Complex Exponential Map

Abstract

In the framework of ergodic optimization for complex dynamical systems, determining the extremal Lyapunov exponents of periodic orbits remains a significant challenge. This paper fully resolves the problem of how weakly repelling a periodic orbit of the complex exponential map $z \mapsto e^z$ can be. We rigorously prove that for any periodic orbit, its multiplier $\lambda$ satisfies the absolute lower bound $\vert{}\lambda\vert{} \ge \theta$, where $\theta \approx 1.3745570107$. Furthermore, we establish that this minimal bound is exactly attained at the principal fixed points $L^*$ and $\overline{L^*}$, representing the Lyapunov ground state of the system. To prove this result, we introduce a novel block decomposition technique (the Block Lemma) that effectively controls the dynamical behavior of orbits traversing the left half-plane. This mechanism allows us to uniformly bound deep excursions ($\text{Re}(z_j) < 0$) relative to the period of the cycle. The analytical framework is completed and verified through certified, rigorous numerical computations, providing a comprehensive proof of the minimal repulsion in the exponential family.