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Self-similarity of the Mandelbrot set for real essentially bounded combinatorics

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  • Let us consider a real quadratic-like germ $f_$∗ which is infinitely renormalizable with tripling essentially bounded combinatorics and consider the lamination given by the hybrid classes in the space of quadratic-like germs, then its holonomy map is shown to be $C^1$ at $f_$∗ if the combinatorics of $f_$∗ satisfies a growth condition. As a consequence, a proof of the self-similarity of the Mandelbrot set for this type of combinatorics is given.
    Mathematics Subject Classification: Primary: 37F10, 37F45; Secondary: 37F20, 37F25.

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