Turning numbers for periodic orbits of disk homeomorphisms

We study braid types of periodic orbits of orientation preserving disk homeomorphisms. If the orbit has period n, we take the closure of the nth power of the corresponding braid and consider linking numbers of the pairs of its components, which we call turning numbers. They are easy to compute and turn out to be very useful in the problem of classification of braid types, especially for small n. This provides us with a simple way of getting useful information about periodic orbits. The method works especially well for disk homeomorphisms that are small perturbations of interval maps.