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Register a new userLocal connectivity of Julia sets for rational maps with Siegel disks
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We prove that a long iteration of rational maps is expansive near boundaries of bounded type Siegel disks. This leads us to extend Petersens local connectivity result on the Julia sets of quadratic Siegel polynomials to a general case.
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Consider a polynomial $f$ of degree $d geq 2$ that has a Siegel disk $Delta_f$ with a rotation number of bounded type. We prove that there does not exist a hedgehog containing $Delta_f$. Moreover, if the Julia set $J_f$ of $f$ is connected, then it is locally connected at the Siegel boundary $partial Delta_f$.
In 1980s, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurstons Theorem.
We show that for each $din (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a emph{finest monotone map $ph$ onto a locally connected continuum $J_{sim_P}$}, i.e. a monotone map $ph:Jto J_{sim_P}$ such that for any other monotone map $psi:Jto J$ there exists a monotone map $h$ with $psi=hcirc ph$. Then we extend $ph$ onto the complex plane $C$ (keeping the same notation) and show that $ph$ monotonically semiconjugates $P|_{C}$ to a emph{topological polynomial $g:Cto C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwis fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $ph$ not to collapse $J$ into a point.
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
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