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 2m – modal degree d type m = 4, d = 2 periodic extrema are attractors; no wandering intervals; no intervals of periodic points; periodic attractors are essential. 1 0 1 2 3 0 G G mod 1

5. Blaschke Products Control of position of critical values

6. Theorem:

7. The family is full Theorem:

8. History 0 1

12. Parameter space i) ii)

13. Realization of finite combinatorics 3) The combinatorics

15. The map of critical values