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Consider a quadratic polynomial with a fixed Siegel disc of bounded type. Using an adaptation of complex a priori bounds for critical circle maps, we prove that this Siegel polynomial is conformally mateable with the basilica polynomial.
In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $mathcal{F}_a$: $$left(frac{aw-1}{w-1}right)^2+left(frac{aw-1}{w-1}right)left(frac{az+1}{z+1}right) +left(frac{az+1}{z+1}right)^2=3$$
We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}
We reconsider the Schroder-Siegel problem of conjugating an analytic map in $mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Brunos one on th
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.
Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show that there are only two possibilities for the str