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

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

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

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 G G mod 1

Blaschke Products Control of position of critical values

Theorem:

The family is full Theorem:

History 0 1

Parameter space i) ii)

Realization of finite combinatorics 3) The combinatorics

The map of critical values