# American Institute of Mathematical Sciences

March  2004, 11(2&3): 241-260. doi: 10.3934/dcds.2004.11.241

## External arguments and invariant measures for the quadratic family

 1 División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco, AP. 24, Cunduacán Tabasco, 86690, Mexico

Received  December 2002 Revised  May 2004 Published  June 2004

There is a correspondence between the boundary of the main hyperbolic component $W_0$ of the Mandelbrot set $M$ and $M \cap \mathbb R$ . It is induced by the map $T(\theta)=1/2+\theta/4$ defined on the set of external arguments of $W_0$.
If $c$ is a point of the boundary of $W_0$ with internal argument $\gamma$ and external argument $\theta$ then $T(\theta)$ is an external argument of the real parameter $c'\in M.$ We give a characterization, for the parameter $c'$ corresponding to $\gamma$ rational, in terms of the Hubbard trees. If $\gamma$ is irrational, we prove that $P_{c'}$ does not satisfy the $CE$ condition. We obtain an asymmetrical diophantine condition implying the existence of an absolutely continuous invariant measure (a.c.i.m.) for $P_{c'}$. We also show an arithmetic condition on $\gamma$ preventing the existence of an a.c.i.m.
Citation: Gamaliel Blé. External arguments and invariant measures for the quadratic family. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 241-260. doi: 10.3934/dcds.2004.11.241
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