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A Generalized Mandelbrot Set of Polynomials of Type πΈπ for Bicomplex Numbers
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Ahmad Zireh
Published/Copyright:
March 10, 2010
Abstract
We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type πΈπ, ππ(π€) = π€(π€ + π)π. Rochon [Fractals 8: 355β368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form π€2 + π is connected. We prove that our generalized Mandelbrot set of polynomials of type πΈπ, ππ(π€) = π€(π€ + π)π, is connected.
Key words and phrases:: Bicomplex Numbers; Mandelbrot set; Connectedness locus; polynomial dynamics; DouadyβHubbard's theorem
Received: 2006-12-04
Published Online: 2010-03-10
Published in Print: 2008-March
Β© Heldermann Verlag
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Keywords for this article
Bicomplex Numbers;
Mandelbrot set;
Connectedness locus;
polynomial dynamics;
DouadyβHubbard's theorem
Articles in the same Issue
- Functional Equations and ΞΌ-Spherical Functions
- On the Uniqueness of Meromorphic Functions That Share Two Sets
- Convergence of the Ishikawa Iterates for Multi-Valued Mappings in Convex Metric Spaces
- The Existence of Solutions for Nonlinear Operator Equations
- A Measure Theoretical Version of the Aleksandrov Theorem
- Coincidence and Fixed Points of Contractive Type Multivalued Maps
- On the πΏ1-Convergence of Modified Cosine Sums
- On the Approximation Properties of a Class of Convolution Type Nonlinear Singular Integral Operators
- A Metric Deformation to Obtain a Positive/Negative Gaussian Curvature on a Disk
- On Automorphism Groups of Ο-Trees
- Growth and Approximation of Entire Harmonic Functions in π π, π > 3
- On Some Properties of the Dirichlet Problem at Resonance
- GrΓΆbner Bases for the Modules Over Noetherian Polynomial Commutative Rings
- Singular 2D Behaviors: Homologies
- Integrating Over the Inverse Image of Functions in the Besov Space
- The Fourth Order of Accuracy Decomposition Scheme for Abstract Hyperbolic Equation
- On a Generalization of Partial Isometries in Banach Spaces
- A Generalized Mandelbrot Set of Polynomials of Type πΈπ for Bicomplex Numbers