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Register a new userPrimitive tuning via quasiconformal surgery
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Using quasiconformal surgery, we prove that any primitive, postcritically-finite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable.
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We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-circles but the Julia set is non-locally connected.
Let $f_0$ be a polynomial of degree $d_1+d_2$ with a periodic critical point $0$ of multiplicity $d_1-1$ and a Julia critical point of multiplicity $d_2$. We show that if $f_0$ is primitive, free of neutral periodic points and non-renormalizable at the Julia critical point, then the straightening map $chi_{f_0}:mathcal C(lambda_{f_0}) to mathcal C_{d_1}$ is a bijection. More precisely, $f^{m_0}$ has a polynomial-like restriction which is hybrid equivalent to some polynomial in $mathcal C_{d_1}$ for each map $f in mathcal C(lambda_{f_0})$, where $m_0$ is the period of $0$ under $f_0$. On the other hand, $f_0$ can be tuned with any polynomial $gin mathcal C_{d_1}$. As a consequence, we conclude that the straightening map $chi_{f_0}$ is a homeomorphism from $mathcal C(lambda_{f_0})$ onto the Mandelbrot set when $d_1=2$. This together with the main result in [SW] solve the problem for primitive tuning for cubic polynomials with connected Julia sets thoroughly.
We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-preserving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue measure.
In this article, we give a quasi-final classification of quasiconformal Anosov flows. We deduce a very interesting differentable rigidity result for the orbit foliations of hyperbolic manifold of dimension at least three.
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.
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