The geometry of the Weil-Petersson metric in complex dynamics

TL;DR

This paper explores the Weil-Petersson metric in complex dynamics, specifically on the space of degree 2 Blaschke products, analyzing its geometric properties, boundary behavior, and decay rates during degenerations.

Contribution

It introduces a new analogue of the Weil-Petersson metric for Blaschke products, investigates its metric completion, and estimates its decay along degenerations.

Findings

The metric completion includes geometrically finite parameters on the boundary.

Bounds are established for intersections with invariant subsets near the boundary.

The decay rate of the Weil-Petersson metric along radial degenerations is computed.

Abstract

In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion. For the upper bound, we estimate the intersection of a circle $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations.