On the parameter space of fibered hyperbolic polynomials

TL;DR

This paper applies quasiconformal surgery to fibered hyperbolic polynomials with quasiperiodic forcing, demonstrating that the fibered multiplier map is a holomorphic submersion, extending classical dynamics concepts to a fibered setting.

Contribution

It introduces a quasiconformal surgery technique for fibered maps and proves the fibered multiplier map is a holomorphic submersion in this context.

Findings

Fibered multiplier map is a holomorphic submersion

Extension of classical multiplier change to fibered hyperbolic polynomials

Application to quadratic maps with Diophantine frequency

Abstract

We present an application of quasiconformal (QC) surgery for holomorphic maps fibered over an irrational rotation of the unit circle, also known as quasiperiodically forced (QPF) maps. It consists of modifying the fibered multiplier of an attracting invariant curve for a QPF hyperbolic polynomial. This is the analogue of the classical change of multiplier of an attracting cycle in the one-dimensional iteration case, to parametrize hyperbolic components of the Mandelbrot set. Our goal is to show that, for a family of QPF quadratic maps with Diophantine frequency, the fibered multiplier map associated to its unique attracting invariant curve, as a complex counterpart of the Lyapunov exponent, is a holomorphic submersion on a complex Banach manifold.