Uniform expansivity outside the critical neighborhood in the quadratic family

TL;DR

This paper employs rigorous numerical methods to establish lower bounds on the expansivity exponent outside the critical neighborhood in the quadratic family, providing insights into parameter distribution and ensuring mathematical rigor.

Contribution

It introduces a computational approach to rigorously estimate expansivity exponents across parameter intervals in the quadratic family, including the critical neighborhood radius.

Findings

Computed lower bounds for expansivity exponents for many parameter intervals.

Identified distribution patterns of critical neighborhood radii and exponents.

Provided publicly available software and results for reproducibility.

Abstract

We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We compute a possibly small radius of the critical neighborhood, and a lower bound for the corresponding expansivity exponent outside this neighborhood, valid for all the parameters in each of the intervals. We illustrate and study the distribution of the radii and these exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/..