The dynamical degrees of rational surface automorphisms
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- by Kyounghee Kim;
- Conform. Geom. Dyn. 28 (2024), 62-87
- DOI: https://doi.org/10.1090/ecgd/392
- Published electronically: May 31, 2024
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Abstract:
The induced action on the Picard group of a rational surface automorphism with positive entropy can be identified with an element of the Coxeter group associated to $E_n, n\ge 10$ diagram. It follows that the set of dynamical degrees of rational surface automorphisms is a subset of the spectral radii of elements in the Coxeter group. This article concerns the realizability of an element of the Coxeter group as an automorphism on a rational surface with an irreducible reduced anticanonical curve. For any unrealizable element, we explicitly construct a realizable element with the same spectral radius. Hence, we show that the set of dynamical degrees and the set of spectral radii of the Coxeter group are, in fact, identical. This has been shown by Uehara [Ann. Inst. Fourier (Grenoble) 66 (2016), pp. 377–432] by explicitly constructing a rational surface automorphism. This construction depends on a decomposition of an element of the Coxeter group. Our proof is conceptual and provides a simple description of elements of the Coxeter group, which are realized by automorphisms on anticanonical rational surfaces.References
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Bibliographic Information
- Kyounghee Kim
- Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
- Email: kim@math.fsu.edu
- Received by editor(s): July 31, 2023
- Received by editor(s) in revised form: March 14, 2024
- Published electronically: May 31, 2024
- Additional Notes: This work was partially supported by the National Science Foundation under DMS-1928930 while the author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2022 semester
- © Copyright 2024 American Mathematical Society
- Journal: Conform. Geom. Dyn. 28 (2024), 62-87
- MSC (2020): Primary 37F15, 14E07
- DOI: https://doi.org/10.1090/ecgd/392
- MathSciNet review: 4753260