Y. Jiang (CUNY) Dual Derivatives And Teichmüller Structures for
Dynamical Systems
In this talk we will use a simple example $z \to z^2$ of circle maps to introduce the symbolic model of all degree two analytic expanding circle maps. Following this is that any two degree two circle analytic expanding maps are topologically conjugate. Furthermore, we will show that the conjugacy is actually quasisymmetric. Using this property, we will introduce a Teichmüller structure over all degree two analytic circle expanding maps. This Teichmüller structure is not complete. However, we will study its completion. We will give a geometric description of this completion. First, we will define a uniformly symmetric degree two circle map. Again, we will show that any two such maps are quasisymmetrically conjugate. The Teichmüller structure over all uniformly symmetric degree two circle maps will be the completion. A uniformly symmetric degree two circle map may be singular in the sense that it may map a positive Lebesgue measurable set into a zero Lebesgue measurable set. Therefore, it may not be differentiable at all. But we will define the dual derivative of a uniformly symmetric circle map. We will prove that the dual derivative always exists and is continuous on the dual symbolic space (dual Cantor set). Two conditions are discovered for the dual derivative of a uniformly symmetric circle map. We show that a continuous function on the dual Cantor set is the dual derivative of a uniformly symmetric circle map if and only if it satisfies these two conditions. Using this result we set up another model of the Teichmüller space of all uniformly symmetric degree two circle maps. The linear models of a uniformly symmetric circle map will be also discussed and used as another model of the Teichmüller space.
