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

• Main Authors: Pavel Trojovský, K Venkatachalam
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Fractal and Fractional
• Subjects: fractal; Mandelbrot set; Julia set; Möbius transformation; iterated function; Catalan numbers
• Online Access: https://www.mdpi.com/2504-3110/5/3/92

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mi>λ</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>λ</mi></mrow></semantics></math></inline-formula>. Their generalization was based on the composition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>η</mi><mi>λ</mi></msub></semantics></math></inline-formula> with the Möbius transformation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>z</mi></mfrac></mrow></semantics></math></inline-formula> at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>(</mo><msub><mi>η</mi><mi>λ</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> (at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.