On invariant line

We show that a rational function of degree 2 admits an invariant line eld with respect to some measure , which is an equilibrium state of a HH older continuous potential whose topological pressure is greater than its supremum, only in very special cases when the Julia set is either a geometric circle or an interval or it is totally disconnected and contained in a real-analytic curve. Let f : C I ! C I be a rational function of degree 2 and let : J(f) ! IR be a HH older continuous potential deened on the Julia set such that P() > sup(), where P() is the topological pressure of with respect to the map f : J(f) ! J(f). For the deenition and various properties of topological pressure the reader may consult Bo], Ru], Wa] or PU] for example. It is true (see the same sources as above) that P() has the following theoretical-metric characterization called the variational principle. P() = supfh (f) + Z ddg; where the supremum is taken over all Borel probability f-invariant measures supported on J(f) and h (f) is the metric entropy of f with respect to the measure. A Borel probability f-invariant measure on J(f) is said to be an equilibrium stste for if h (f)+ R dd = P(). On knows that for each continuous potential on J(f) there exists at least one equilibrium state; this is due to M. Lyubich in Ly]. For HH older continuous potentials which satisfy the condition P() > sup(), the equilibrium state is in fact unique, as shown in DU] (see also Pr]). We denote this unique equilibrium state by. It is ergodic; further dynamical and ergodic properties, including metric exactness and the Central Limit Theorem have been established in DU], Pr], DPU] and Ha]. The main result of this paper is contained in the following.