The most fundamental dynamical invariants of a rational mapping $f:X\to X$ are its dynamical degrees, which describe the rate of growth for the action of iterates of $f$ on the cohomology of $X$. When $f$ has non-empty indeterminacy set, these quantities can be very difficult to determine, especially the dynamical degrees for the action on $H^{(p,p)}(X)$ for $p > 1$. The primary examples in which they are known are monomial maps and certain psuedo-automorphism of 3-dimensional manifolds.

In this work, we study rational mappings of the Deligne-Mumford compactification of the moduli space of $n$ points on the Riemann sphere. Because of the beautiful recursive structure of this space, we are able to provide many new examples of rational maps of algebraic varieties of arbitrary dimension for which the dynamical degrees can be computed. We conclude with several specific examples. This is joint work with Sarah Koch.