Infinite-dimensional Thurston theory and transcendental dynamics IV: dependence on parameters and escape on (pre-)periodic rays

TL;DR

This paper studies how transcendental entire functions, composed of polynomials and exponentials with escaping singular values, depend on parameters, extending previous classifications to include pre-periodic rays using continuity arguments.

Contribution

It generalizes the classification of such functions to cases with pre-periodic rays, enhancing understanding of their parameter dependence and dynamic behavior.

Findings

Extended classification to pre-periodic rays

Demonstrated parameter dependence via continuity arguments

Analyzed escape dynamics on (pre-)periodic rays

Abstract

We consider transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. Based on their classification in [B3] we investigate their dependence on parameters, that is, on the potentials and external addresses of singular values. Using continuity argument we generalize the classification in [B3] to the case when dynamic rays containing singular values are (pre-)periodic.