Discontinuity of Straightening in Anti-holomorphic Dynamics: I

Discontinuity of straightening in anti-holomorphic dynamics: II

In [M3], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci, and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.

Combinatorics and topology of straightening maps II: Discontinuity

We continue the study of straightening maps for the family of polynomials of degree $d \ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and self-similarity of the Mandelbrot set. In the quadratic case, straightening maps are always continuous, and this is one of the critical steps to prove the Mandelbrot set has small copies in itself. On the other hand, for higher degree case, we do not have such a nice self-similar property: As expected from an example of a cubic-like family with discontinuou…

Schwarz reflections and anti-holomorphic correspondences

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of…

Dynamics of asymptotically holomorphic polynomial-like maps

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

Self-similarity for the tricorn

We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author, we show that there are many homeomorphic copies of the Mandelbrot sets. With help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a "baby tricorn" centered at the airplane, hence not a homeomorphism.

关于分段多项式全纯系统极限环数量的研究

On the number of limit cycles for piecewise polynomial holomorphic systems

本文研究带直线不连续的分段多项式全纯系统极限环下界，综合平均函数零点、退化Andronov-Hopf分支和滑移效应，并利用Poincaré-Miranda定理构造出具有3个极限环的显式分段线性全纯系统，改进了文献中仅1个极限环的已知结果。