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Classification Of Fatou Components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou. Rational case If f is a rational function :f = \frac defined in the extended complex plane, and if it is a nonlinear function (degree > 1) : d(f) = \max(\deg(P),\, \deg(Q))\geq 2, then for a periodic component U of the Fatou set, exactly one of the following holds: # U contains an attracting periodic point # U is parabolic # U is a Siegel disc: a simply connected Fatou component on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. # U is a Herman ring: a double connected Fatou component (an annulus) on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle. File:Julia-set_N_z3-1.png, Julia set (white) and Fatou set (dark red/green/blue) for f: z\mapsto z-\frac(z) with g: z \mapsto z^3-1 in the complex plane. Basilica Juli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
