Computing dynamical degrees

TL;DR

This paper develops methods to compute dynamical degrees of rational maps on certain moduli spaces, providing explicit calculations in low dimensions and revealing their functorial behavior under iteration.

Contribution

It introduces a stratified approach to compute dynamical degrees for rational maps on Deligne-Mumford compactifications, with explicit results in dimensions up to 3 and first degrees up to dimension 5.

Findings

Explicit computation of all dynamical degrees for maps on spaces of dimension ≤ 3.

First dynamical degrees computed for maps on spaces of dimension ≤ 5.

Demonstration of functorial behavior of cohomological action under iteration.

Abstract

The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can be very difficult to determine. We study rational maps $f:X^N\dashrightarrow X^N$, where $X^N$ is isomorphic to the Deligne-Mumford compactification $\overline {\mathcal M}_{0,N+3}$. We exploit the stratified structure of $X^N$ to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this article, we explicitly compute all of the dynamical degrees for all such maps $f:X^N\dashrightarrow X^N$, where $\mathrm{dim}(X^N)\leq 3$ and the first dynamical degrees for the mappings where $\mathrm{dim}(X^N)\leq 5$. These examples naturally arise in the setting of Thurston's topological characterization of rational maps.