How the periodic points, point orbits and orbits of subvarieties of a meromorphic self-map or multi-valued self-correspondence on a compact Kahler manifold spread out as the iteration order grows. A student learns why the count of isolated periodic points is bounded by the algebraic entropy (the largest dynamical degree, read off the action of iterates on Hodge cohomology), and how normalising the current of integration on the graph of the n-th iterate and intersecting it with the diagonal turns that count into a canonical invariant probability measure. The machinery for making those intersections legitimate in arbitrary bidegree -- super-potentials of positive closed currents and the tangent-class theory of densities that detects dimension excess -- is developed alongside, and applied to holomorphic endomorphisms of projective space, Henon-type polynomial automorphisms, surface automorphisms and modular correspondences.
A survey of how orbits, subvarieties and periodic points distribute themselves under a dominant meromorphic self-map or multi-valued self-correspondence f of a compact Kahler manifold X of dimension …