Julia Sets: Periodic and Non-Periodic Points Explained

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SUMMARY

Julia sets are defined as the closure of all repelling periodic points of a complex map f. However, they inherently contain both periodic and non-periodic points, which can lead to confusion. The distinction lies in understanding that the Julia set is not limited to periodic points alone; it encompasses the closure of its own set, integrating non-periodic points as well. This duality is crucial for comprehending the full structure of Julia sets in complex dynamics.

PREREQUISITES
  • Understanding of complex maps in mathematics
  • Familiarity with the concept of periodic points
  • Knowledge of closure properties in topology
  • Basic grasp of dynamical systems
NEXT STEPS
  • Research the properties of complex maps and their dynamics
  • Explore the role of periodic points in dynamical systems
  • Study the closure operation in topological spaces
  • Investigate the differences between periodic and non-periodic points in Julia sets
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Mathematicians, students of complex dynamics, and anyone interested in the properties of Julia sets and their implications in dynamical systems.

Kortirion
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I'm a little bewildered when reading about these Julia sets. From the definition a Julia set is the closure of all repelling periodic points of a complex map f.
However I read that a Julia set always contain periodic and non-periodic points. Wasn't the definition including only periodic points? What am I missing here.
 
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The "Julia set" in the second definition is the closure of the "Julia set" of the first definition.
 

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