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3 results on '"Julia set"'

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

Author

Trojovský, Pavel and Venkatachalam, K.

Subjects
*ITERATIVE methods (Mathematics), *LOGICAL prediction, *COEFFICIENTS (Statistics), *BINOMIAL coefficients, *MATHEMATICAL analysis
Abstract

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

Published
2021
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2. The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

Author

Pavel Trojovský and K Venkatachalam

Subjects
fractal, Mandelbrot set, Julia set, Möbius transformation, iterated function, Catalan numbers, Thermodynamics, QC310.15-319, Mathematics, QA1-939, Analysis, QA299.6-433
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.

Published
2021
Full Text
View/download PDF

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

Author

K. Venkatachalam and Pavel Trojovský

Subjects
Statistics and Probability, Generalization, MathematicsofComputing_GENERAL, Julia set, Mandelbrot set, iterated function, Combinatorics, Catalan number, fractal, QA1-939, Mathematics, QA299.6-433, Conjecture, Series (mathematics), Möbius transformation, Statistical and Nonlinear Physics, Function (mathematics), TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, Iterated function, Thermodynamics, Catalan numbers, QC310.15-319, Analysis
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.

Published
2021

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