Rational functions with real multipliers
Alexandre Eremenko, Sebastian van Strien
TL;DR
This paper investigates rational functions with real multipliers at repelling periodic points, proving their Julia sets lie on a circle, and characterizes functions with Julia sets on circles.
Contribution
It establishes that rational functions with all repelling periodic points having real multipliers have Julia sets contained in a circle, linking multiplier properties to geometric Julia set structure.
Findings
Julia set of such functions is contained in a circle
If Julia set is on a smooth curve, it is on a circle
Characterization of rational functions with Julia sets on circles
Abstract
Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.
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Taxonomy
TopicsMathematical and Theoretical Analysis · Numerical Methods and Algorithms · Optimization and Variational Analysis