External arguments and invariant measures for the quadratic family (Q1884221)

Here, the quadratic family \(P_c(z) =z^2 +c\) is studied. There is a corrspondence 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 \(T(\vartheta)= 1/2+ \vartheta/4\) 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 \(\vartheta\) then \(T(\vartheta)\) is an external argument of the real parameter \(c' \in M\). We give a characterization, for the parameter \(c'\) corresponding to \(\gamma\) rational, in term of the Hubbard trees. If \(\gamma\) is irrational, we prove that \(P_{c'}\) does not satisty the CE condition. We obtain an asymmetrical Diophantine condition implying the existence of an absolutely continuous invariant measure for \(P_{c'}\). The paper is well written and organized.