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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