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Visualization of Mandelbrot and Julia Sets of Möbius Transformations
- Source :
- Fractal and Fractional, Vol 5, Iss 73, p 73 (2021), Fractal and Fractional, Volume 5, Issue 3
- Publication Year :
- 2021
- Publisher :
- MDPI AG, 2021.
-
Abstract
- This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.
- Subjects :
- Statistics and Probability
Pure mathematics
Mathematics::Dynamical Systems
Generalization
Hyperbolic geometry
Julia set
02 engineering and technology
Mandelbrot set
01 natural sciences
hyperbolic geometry
symbols.namesake
Fractal
fractal
0202 electrical engineering, electronic engineering, information engineering
QA1-939
0101 mathematics
Mathematics
Möbius transformation
QA299.6-433
010102 general mathematics
Statistical and Nonlinear Physics
Function (mathematics)
symbols
Thermodynamics
020201 artificial intelligence & image processing
Affine transformation
QC310.15-319
Analysis
Subjects
Details
- Language :
- English
- ISSN :
- 25043110
- Volume :
- 5
- Issue :
- 73
- Database :
- OpenAIRE
- Journal :
- Fractal and Fractional
- Accession number :
- edsair.doi.dedup.....d288e57d730abedcb1d657719df82b3b