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St. Petersburg Seminar on Representation Theory and Dynamical Systems
May 23, 2012 15:00, St. Petersburg, PDMI, room 311 (nab. r. Fontanki, 27)
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Julia sets converging to filled quadratic Julia sets
R. Kozma Stony Brook University
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Abstract:
Previous results by Devaney et al. have shown that for the family of singularly perturbed quadratic maps $z^2 + \lambda/z^2$ the Julia sets converge to the unit disk as $\lambda \to 0$. We give a generalization
of this result to maps of the family
$$
F_\lambda(z) = z^2 + c +\lambda/z^2
$$ where $c$ is the center of a hyperbolic component of the
Mandelbrot set. Using symbolic dynamics and Cantor necklaces, we show
that as $\lambda \to 0$, the Julia set of $F_\lambda$ converges to the
filled Julia set of $z^2+c$.
Language: English
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