The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

Pavel Trojovský,K Venkatachalam

doi:10.3390/fractalfract5030092

Abstract

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

Highlights

We denote by η j the j-th iterate of η, that is, j−times z
The filled-in Julia set of η is defined as
We show the important connection of the sequencek≥0 to the Catalan numbers

Summary

Introduction

Citation: Trojovský, P.; Venkatachalam, K. The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-MöbiusIterated Function. Fractal Fract. 2021, 5, 92. https://doi.org/Let η : C → C be a monic complex polynomial of degree d ≥ 2. We denote by η j the j-th iterate of η, that is, j−times z }| {j η (z) = η (η (η (· · · η (z) · · · ))).The filled-in Julia set of η is defined asK (η ) = {z ∈ C : η j (z) does not diverge}10.3390/fractalfract5030092Academic Editor: Michel L. Lapidus

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