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Register a new userRenormalization in the Golden-Mean Semi-Siegel Henon Family: Non-Quasisymmetry
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Jonguk Yang
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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.
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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.
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.
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.
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.
It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative H{e}non map. Special numerical method of estimation of the Siegel disk scaling center position (for 1D maps it corresponds to extremum) for multi-dimensional invertible maps are developed.
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