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

Abstract

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function eta(lambda)(z)=z(2)+lambda. Their generalization was based on the composition of eta(lambda) with the Mobius transformation mu(z)=1/z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of mu(eta(lambda)(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.