2014/03/24 by Sarah Koch, Koch, Sarah, Roland K. W. Roeder +1
Mathematics · #32H50 #37F99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1403.5840
openalex publication_date 2014/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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:XN\dashrightarrow XN, where XN is isomorphic to the Deligne-Mumford compactification \mathcal M0,N+3. We exploit the stratified structure of XN 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:XN\dashrightarrow XN, where dim(XN)≤ 3 and the first dynamical degrees for the mappings where dim(XN)≤ 5. These examples naturally arise in the setting of Thurston's topological characterization of rational maps.