External arguments and invariant measures for the quadratic family

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.