Abstract:
The phase-sensitive experiments on cuprate superconductors have told us about the symmetry of the condensate wavefunction. However, they can not determine the pairing symmetry of Cooper pairs. To describe a superconducting state, two wavefunctions are needed, condensate wavefunction and pairing wavefunction. The former describes the entirety movement of the pairs and the latter describes the relative movement of the two electrons within a pair. The $\pi$-phase shift observed in the phase sensitive Josephson measurements can not prove that the pairing state is d-wave. We present here a new explanation and predict some new observable phenomena.

Abstract:
By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points.

Abstract:
The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichm\"uller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichm\"uller structure. Moreover, this Teichm\"uller structure is a complete structure and is the completion of the space of smooth potentials under this Teichm\"uller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.

Abstract:
We introduce a function model for the Teichm\"uller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichm\"uller space. We prove that the identity map from the Teichm\"uller space equipped with the usual Teichm\"uller metric to the Teichm\"uller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichm\"uller space equipped with this new metric to the Teichm\"uller space equipped with the usual Teichm\"uller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichm\"uller metric on the Teichm\"uller space. We give a remark about the pressure metric and the Weil-Petersson metric.

Abstract:
We construct a subset of the Mandelbrot set which is dense on the boundary of the Mandelbrot set and which consists of only infinitely renormalizable points such that the Mandelbrot set is locally connected at every point of this subset. We prove the local connectivity by finding bases of connected neighborhoods directly.

Abstract:
The term integrable asymptotically conformal at a point for a quasiconformal map defined on a domain is defined. Furthermore, we prove that there is a normal form for this kind attracting or repelling or super-attracting fixed point with the control condition under a quasiconformal change of coordinate which is also asymptotically conformal at this fixed point. The change of coordinate is essentially unique. These results generalize K\"onig's Theorem and B\"ottcher's Theorem in classical complex analysis. The idea in proofs is new and uses holomorphic motion theory and provides a new understanding of the inside mechanism of these two famous theorems too.

Abstract:
Geometric models and Teichm\"uller structures have been introduced for the space of smooth expanding circle endomorphisms and for the space of uniformly symmetric circle endomorphisms. The latter one is the completion of the previous one under the Techm\"uller metric. Moreover, the spaces of geometric models as well as the Teichm\"uller spaces can be described as the space of H\"older continuous scaling functions and the space of continuous scaling functions on the dual symbolic space. The characterizations of these scaling functions have been also investigated. The Gibbs measure theory and the dual Gibbs measure theory for smooth expanding circle dynamics have been viewed from the geometric point of view. However, for uniformly symmetric circle dynamics, an appropriate Gibbs measure theory is unavailable, but a dual Gibbs type measure theory has been developed for the uniformly symmetric case. This development extends the dual Gibbs measure theory for the smooth case from the geometric point of view. In this survey article, We give a review of these developments which combines ideas and techniques from dynamical systems, quasiconformal mapping theory, and Teichm\"uller theory. There is a measure-theoretical version which is called $g$-measure theory and which corresponds to the dual geometric Gibbs type measure theory. We briefly review it too.

Abstract:
We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as $\varepsilon $ goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed $\sqrt {\varepsilon}$. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting mapping with respect to the Ulam-von Neumann type metric.

Abstract:
We give a brief review of holomorphic motions and its relation with quasiconformal mapping theory. Furthermore, we apply the holomorphic motions to give new proofs of famous Konig's Theorem and Bottcher's Theorem in classical complex analysis.

Abstract:
We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the critical points of a geometrically finite one-dimensional mapping form a complete set of $C^{1}$-invariants within a topological conjugacy class.