We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval $I$ affects the dynamics of an associated many-to-one skew product map of the cylinder $(\mathbb{R}/\mathbb{Z})\times I$.

We consider a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$, and prove the following Quasi-Additivity Law: If the total extremal width $\sum \mathcal{W}(S\smallsetminus D_i)$ is big enough (depending on $N$) then it is comparable with the extremal width $\mathcal{W} (S,\cup D_i)$ (under a certain ``separation assumption'') . We also consider a branched covering $f: U\rightarrow V$ of degree $N$ between two disks that restricts to a map $\Lambda\rightarrow B$ of degree $d$ on some disk $\Lambda \Subset U$. We derive from the Quasi-Additivity Law that if $\mod(U\smallsetminus \Lambda)$ is sufficiently small, then (under a ``collar assumption'') the modulus is quasi-invariant under $f$, namely $\mod(V\smallsetminus B)$ is comparable with $d^2 \mod(U\smallsetminus \Lambda)$. This Covering Lemma has important consequences in holomorphic dynamics which will be addressed in the forthcoming notes.

This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the parameters for which critical orbits are periodic, and I called such curves left and right bones. Using quasiconformal surgery methods and rigidity I showed that the bones are simple smooth arcs that join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of the maps evolves along the bones. The behavior of the topological entropy function of the polynomials in my family is closely related to the structure of the bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.

The punctured solenoid $\mathcal{H}$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $\mathcal{H}$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $\mathcal{H}$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $\mathcal{H}$ giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of $\mathcal{H}$, which is presumably the main result of this paper. Moreover, each word in these generators admits a normal form, and the natural equivalence relation on normal forms is described. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of $\mathcal{H}$. All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.