We investigate the local behaviour of branched covering maps at their branching points and introduce a notion of a branched derivative, similar to a derivative for diffeomorphisms. Then, under an additional assumption that the map is locally area preserving, we look at the dynamics in a neighbourhood of a periodic branching point. The two stable (hyperbolic) cases are similar to the usual picture at a hyperbolic periodic point, with a few important differences. In particular, in the case analogous to saddle behaviour, one gets one expanding direction and a Cantor set of contracting directions. © 2005 IOP Publishing Ltd and London Mathematical Society.
