On pinching deformations of rational maps

We introduce the notion of the dynamical length of an invariant arc of a rational map R. A pinching deformation is a sequence of topological deformations of R such that the corresponding dynamical length shrinks to zero. We show that if the sequence converges to a rational map then the spherical diameter of the corresponding arc also shrinks to zero. We use this result to show that if the grand orbits of the closure of finitely many such arcs separate the Julia set, the deformations of R diverge. This is a generalization of a result stated by P. Makienko but with a different approach. We also present a rich collection of examples.