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Register a new userThe boundaries of golden-mean Siegel disks in the complex quadratic Henon family are not smooth
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As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic Henon maps are bounded by topological circles. In this paper we investigate the geometric properties of such curves, and demonstrate that they cannot be $C^1$-smooth.
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For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic Henon maps of two complex variables.
It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Henon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Henon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Henon map is non-rigid.
Let $f$ be a polynomial map of the Riemann sphere of degree at least two. We prove that if $f$ has a Siegel disk $G$ on which the rotation number satisfies a diophantine condition, then the boundary of $G$ contains a critical point.
We show that the dynamics of sufficiently dissipative semi-Siegel complex Henon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_delta$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.
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 $partial Delta_f$.
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