Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Let K be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over K. Let f : X proves X and g :Y proves Y be dominant correspondences, and pi : X -> Y a dominant rational map which semi-conjugate f and g, i.e. so that pi circle f = g circle pi. We define relative dynamical degrees lambda(p)(f\pi) >= 1 for any p = 0,..., dim(X) - dim(Y). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy (phi, psi) from pi(2) : (X-2, f(2)) -> (Y-2, g(2)) to pi(1) : (X-1, f(1)) -> (Y-1,g(1)) we have lambda(p)(f(1)vertical bar pi(1)) >= lambda p(f(2)vertical bar pi(2)) for all p. Many of our results are new even when K = C. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.