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Full Families of Multimodal Maps on the Circle

Explore dynamical behaviors of multimodal maps on the circle, detailing models and behaviors with specific characteristics. This study focuses on G(x) mod 1, lift of continuous maps, periodic attractors, Blaschke Products, and control of critical values. Theorems in the family of multimodal maps are also discussed.

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Full Families of Multimodal Maps on the Circle

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  1. Full Families of Multimodal Maps on the Circle E. de Faria, W. de Melo, P. Salomão, E. Vargas

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

  3. Multimodal maps on the circle G(x) mod 1 G(x+1) = G(x) +d d=2

  4. Multimodal maps on the circle G a lift of a continuous map g on 3 • 2m – modal • degree d • type 2 G 1 • periodic extrema are attractors; • no wandering intervals; • no intervals of periodic points; • periodic attractors are essential. 0 -1 1 0 m = 4, d = 2 G mod 1

  5. Blaschke Products Control of position of critical values

  6. Theorem:

  7. The family is full Theorem:

  8. History 1 0

  9. Parameter space ii) i)

  10. Realization of finite combinatorics 3) The combinatorics

  11. The map of critical values

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