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A rational map with infinitely many points of distinct arithmetic degrees

Part of: Arithmetic and non-Archimedean dynamical systems Birational geometry

Published online by Cambridge University Press:  12 April 2019

JOHN LESIEUTRE Open the ORCID record for JOHN LESIEUTRE [Opens in a new window]  and
MATTHEW SATRIANO
JOHN LESIEUTRE
Affiliation:
Penn State University Mathematics Department, 204 McAllister Building, University Park, State College, PA16802, USA email jdl@psu.edu
MATTHEW SATRIANO
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, CanadaN2L 3G1 email msatrian@uwaterloo.ca
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Abstract

Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$. For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$-orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$, which measures the growth rate of the heights of the points $f^{n}(P)$. Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode[STIX]{x1D6FC}_{f}(P)$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $X=\mathbb{P}^{4}$.

Keywords

arithmetic dynamicsalgebraic dynamicsCremona transformations

MSC classification

Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 14E07: Birational automorphisms, Cremona group and generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 3051 - 3055
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Copyright
© Cambridge University Press, 2019

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