Maps with no a priori bounds

TL;DR

This paper establishes upper bounds for polynomial-like map moduli in satellite cases, challenging the existence of universal a priori bounds and exploring implications for Julia set geometry and renormalization.

Contribution

It provides the first upper bounds for PL map moduli in satellite cases based solely on relative period and degree, and analyzes their implications.

Findings

Upper bounds depend only on relative period and degree

A priori bounds do not exist for unbounded relative periods in satellite cases

Moduli of annuli tend to zero under certain renormalization conditions

Abstract

The modulus of a polynomial-like (PL) map is an important invariant that controls distortion of the straightening map and, hence, geometry of the corresponding PL Julia set. Lower bounds on the modulus, called complex a priori bounds, are known in a great variety of contexts. For any rational function we complement this by an upper bound for moduli of PL maps in the satellite case that depends only on the relative period and the degree of the PL map. This rules out a priori bounds in the satellite case with unbounded relative periods. We also apply our tools to obtain lower bounds for hyperbolic lengths of geodesics in the infinitely renormalizable case, and to show that moduli of annuli must converge to 0 for a sequence of arbitrary renormalizations, under several conditions all of which are shown to be necessary.