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Arithmetic degrees and
dynamical degrees of endomorphisms on surfaces
Yohsuke Matsuzawa, Kaoru Sano and Takahiro Shibata |
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Vol. 12 (2018), No. 7, 1635–1657
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Abstract |
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For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, if the map is an automorphism, there exists a Zariski dense set of such points with pairwise disjoint orbits. |
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Keywords
arithmetic degree, dynamical degrees, arithmetic dynamics
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Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11G35, 11G50, 37P05, 37P15, 37P30
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Milestones
Received: 20 March 2017
Revised: 5 April 2018
Accepted: 20 June 2018
Published: 27 October 2018
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Authors |
| Yohsuke Matsuzawa | |
| Graduate school of Mathematical
Sciences University of Tokyo Komaba, Tokyo Japan |
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| Kaoru Sano | |
| Department of Mathematics, Faculty
of Science Kyoto University Japan |
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| Takahiro Shibata | |
| Department of Mathematics, Faculty
of Science Kyoto University Japan |
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