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Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties

Part of: Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  17 October 2018

YOHSUKE MATSUZAWA  and
KAORU SANO
YOHSUKE MATSUZAWA
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan email [email protected]
KAORU SANO
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan email [email protected]
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Abstract

We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.

Keywords

arithmeticalgebraic dynamicsHeightssemi-abelian vareitiesdynamical degree

MSC classification

Primary: 37P30: Height functions; Green functions; invariant measures
Secondary: 37P55: Arithmetic dynamics on general algebraic varieties
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1655 - 1672
Copyright
© Cambridge University Press, 2018

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References

Bombieri, E. and Gubler, W.. Heights in Diophantine Geometry. Cambridge University Press, Cambridge, 2007.Google Scholar
Dang, N.-B.. Degrees of iterates of rational maps on normal projective varieties. Preprint, 2017,arXiv:1701.07760.Google Scholar
Diller, J. and Favre, C.. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6) (2001), 1135–1169.Google Scholar
Dinh, T.-C. and Nguyên, V.-A.. Comparison of dynamical degrees for semi-conjugate meromorphic maps. Comment. Math. Helv. 86(4) (2011), 817–840.Google Scholar
Dinh, T.-C. and Sibony, N.. Equidistribution problems in complex dynamics of higher dimension. Internat. J. Math. 28 (2007), 1750057.Google Scholar
Hartshorne, R.. Algebraic Geometry (Graduate Texts in Mathematics, 52). Springer, New York, 1977.Google Scholar
Hindry, M. and Silverman, J. H.. Diophantine Geometry. An Introduction (Graduate Text in Mathematics, 20). Springer, New York, 2000.Google Scholar
Kawaguchi, S. and Silverman, J. H.. Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties. Trans. Amer. Math. Soc. 368 (2016), 5009–5035.Google Scholar
Kawaguchi, S. and Silverman, J. H.. On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. 713 (2016), 21–48.Google Scholar
Kollár, J. and Mori, S.. Birational Geometry of Algebraic Varieties. Cambridge University Press, Cambridge, 1998.Google Scholar
Lin, J.-L.. On the arithmetic dynamics of monomial maps. Ergod. Th. & Dynam. Sys. doi:10.1017/etds. 2018.5. Published online 13 March 2018.Google Scholar
Matsuzawa, Y.. On upper bounds of arithmetic degrees. Preprint, 2016, arXiv:1606.00598v2.Google Scholar
Matsuzawa, Y., Sano, K. and Shibata, T.. Arithmetic degrees and dynamical degrees of endomorphisms on surfaces. Algebra Number Theory to appear.Google Scholar
Mumford, D.. Abelian Varieties (Tata Institute of Fundamental Research Studies in Mathematics, 5). Tata Institute of Fundamental Research, Bombay, 1970.Google Scholar
Sano, K.. The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees. Preprint, 2017, arXiv:1712.07533.Google Scholar
Silverman, J. H.. Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergod. Th. & Dynam. Sys. 34(2) (2014), 647–678.Google Scholar
Silverman, J. H.. Arithmetic and dynamical degrees on abelian varieties. J. Théor. Nombres Bordeaux 29(1) (2017), 151–167.Google Scholar
Truong, T. T.. (Relative) dynamical degrees of rational maps over an algebraic closed field. Preprint, 2015, arXiv:1501.01523v1.Google Scholar
Vojta, P.. Integral points on subvarieties of semiabelian varieties, I. Invent. Math. 126(1) (1996), 133–181.Google Scholar