Mandelbrot sets for fixed template iterations

TL;DR

This paper explores the dynamics of fixed template iterations involving polynomial functions, analyzing how the connectedness locus varies with parameters and revealing semicontinuity properties and counterexamples.

Contribution

It introduces the study of Mandelbrot sets for fixed template iterations with complex polynomials, highlighting semicontinuity behavior and providing counterexamples.

Findings

Connectedness locus depends upper semicontinuously on parameters for most templates.

Lower semicontinuity does not generally hold, demonstrated by a counterexample.

The work extends classical Mandelbrot set concepts to fixed template polynomial iterations.

Abstract

We study the dynamics of template iterations, consisting of arbitrary compositions of functions chosen from a finite set of polynomials. In particular, we focus on templates using complex unicritical maps in the family $\{ z^d + c, c \in \mathbb{C}, d \ge 2 \}$. We examine the dependence on parameters of the connectedness locus for a fixed template and show that, for most templates, the connectedness locus moves upper semicontiuously. On the other hand, one does not in general have lower semicontinuous dependence, and we show this by means of a counterexample.