# American Institute of Mathematical Sciences

December  2006, 16(4): 897-922. doi: 10.3934/dcds.2006.16.897

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

 1 Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, col. Lomas de Chamilpa, C.P. 62210 Cuernavaca, Morelos, Mexico

Received  September 2005 Revised  July 2006 Published  September 2006

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.
Citation: Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
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