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Some Geometric Properties of Julia Sets of Maps
In this work, we will study the geometric properties of Julia sets of the quadratic polynomial maps of the form where is a non-zero complex .
We show that Julia set is the unit circle if c =2 and Julia set is the line segment if c=4 . If 1<z<2.3 .
Then the Julia set is a simple closed curve ,also if 1<z<2.3 then the Julia set is a simple closed curve such that Julia set which contains no smooth arcs, and if then the Julia set is infinitely many different simple closed curves.
In complex dynamics , the iteration theory originated in 1910 [7] . Among the most important concepts in complex dynamics are Julia sets .They were studied by the French mathematician Gaston Julia (1893 – 1978 ) ,
who developed much of theory when he was recovering from his wounds in an army hospital during world war I . He published a long paper in French language in [4],
Julia and Fatou looked at the iteration of the simplest quadratic map of the form ( ) .
In general ,distinct maps have distinct Julia sets , however , there exist distinct polynomial maps , rational maps and entire maps that have the same Julia sets [5], [6].The Julia set of a polynomial typically has a complicated , self – similar structure. Therefore the Julia sets are fractals [2] ,[7] .
However , there exist rational maps whose Julia sets fail to be quasi-self-similar [3] .
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- Julia Sets of Maps
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