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Computing dynamical degrees of rational maps on moduli space

Published online by Cambridge University Press:  21 July 2015

SARAH KOCH  and
ROLAND K. W. ROEDER
SARAH KOCH
Affiliation:
University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109, USA email kochsc@umich.edu
ROLAND K. W. ROEDER
Affiliation:
IUPUI, Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, IN 46202-3267, USA email rroeder@math.iupui.edu
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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 non-empty 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 paper, we explicitly compute all of the dynamical degrees for all such maps $f:X^{N}{\dashrightarrow}X^{N}$, where $\text{dim}(X^{N})\leq 3$ and the first dynamical degrees for the mappings where $\text{dim}(X^{N})\leq 5$. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2538 - 2579
Copyright
© Cambridge University Press, 2015 

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