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Local dynamics of non-invertible maps near normal surface singularities
About this Title
William Gignac and Matteo Ruggiero
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 272, Number 1337
ISBNs: 978-1-4704-4958-2 (print); 978-1-4704-6753-1 (online)
DOI: https://doi.org/10.1090/memo/1337
Published electronically: October 5, 2021
Keywords: Local dynamics,
non-invertible germs,
superattracting germs,
normal surface singularities,
valuation spaces,
b-divisors,
algebraic stability,
attraction rates,
first dynamical degree
Table of Contents
Chapters
- Introduction
- 1. Normal surface singularities, resolutions, and intersection theory
- 2. Normal surface singularities and their valuation spaces
- 3. Log discrepancy, essential skeleta, and special singularities
- 4. Dynamics on valuation spaces
- 5. Dynamics of non-finite germs
- 6. Dynamics of non-invertible finite germs
- 7. Algebraic stability
- 8. Attraction rates
- 9. Examples and remarks
- A. Cusp singularities
Abstract
We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs $f\colon (X,x_0)\to (X,x_0)$, where $X$ is a complex surface having $x_0$ as a normal singularity. We prove that as long as $x_0$ is not a cusp singularity of $X$, then it is possible to find arbitrarily high modifications $\pi \colon X_\pi \to (X,x_0)$ such that the dynamics of $f$ (or more precisely of $f^N$ for $N$ big enough) on $X_\pi$ is algebraically stable. This result is proved by understanding the dynamics induced by $f$ on a space of valuations associated to $X$; in fact, we are able to give a strong classification of all the possible dynamical behaviors of $f$ on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of $f$. Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.- Marco Abate and Jasmin Raissy, Formal Poincaré-Dulac renormalization for holomorphic germs, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1773–1807. MR 3002727, DOI 10.3934/dcds.2013.33.1773
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