Quasiconformal Surgery And Its Applications

As a powerful tool, the original applications of quasiconformal map-pings in complex dynamics is due to Sullivan. Afterwards, under the efforts of Shishikura, many nice results about rational maps by means of quasiconformal surgery, such as the sharp bound of non-repelling cycles and so on was obtained. In fact, there are two fundamental editions about how to establish the conditions of quasiconformal surgery. This paper shows that under some conditions, this two different editions are equivalent, and we make some comparisons between them.This paper mainly discuss the basic principles of quasiconformal surgery and its simple applications. For examples, a polynomial-like map is hybrid conjugate to a poly-nomial, a geometrically attracting fixed point in simply connected Fatou component can be transformed into a superattracting fixed point or irrational indifferent fixed point via surgery, and how to cross-transfer between Siegel discs and Herman rings and so on.Generally speaking, we can describe the process of quasiconformal surgery simply as follows:firstly we combine several dynamics (generally is two) into a quasiregular map with special dynamical properties via "cut and paste", then have the quasiregular map be conjugated to a new rational map using measurable Riemann mapping theorem, up to here, the surgery has completed. But sometimes we are also concerned with the expressions of the new rational maps, not only the dynamical properties.For instance, in this paper, after considering the problems of linearization of ana-lytic functions, we also take into account the problems of analytical linearization near the unit circle making use of quasiconformal surgery. In fact, we have found a family of analytical functions, which possess the properties that they can be linearized near their irrational indifferent fixed points if and only if the rotation number is Brjuno number. At the same time, By the means of quasiconformal surgery, we also find a family of Blashke products R such that the element B of R can be analytical linearized near the unit circle if and only if the rotation number of B on S1 is Brjuno number.The study of the parameter spaces of rational functions is always difficult. In this paper, using quasiconformal surgery, we simply depict the parameter spaces of a family of Blaschke products of degree 3 which possess Herman rings.