Full Families of Multimodal Maps on the Circle

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Full Families of Multimodal Maps on the Circle

• E. de Faria, W. de Melo,
• P. Salomão, E. Vargas

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GOAL

• Find models and parametrize dynamical behaviors.
• Describe each model or behavior.

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Multimodal maps on the circle





G(x) mod 1
G(x1) G(x) d
d2
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Multimodal maps on the circle
G a lift of a continuous map g on





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• 2m modal
• degree d
• type

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G
1

• periodic extrema are attractors
• no wandering intervals
• no intervals of periodic points
• periodic attractors are essential.

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-1
1
0
m 4, d 2
G mod 1
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Blaschke Products
Control of position of critical values
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Theorem
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The family is full
Theorem
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History
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Parameter space
ii)
i)
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Realization of finite combinatorics
3) The combinatorics
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The map of critical values