Arithmetic degrees and dynamical degrees of endomorphisms on surfaces

Matsuzawa, Yohsuke; Sano, Kaoru; Shibata, Takahiro

doi:10.2140/ant.2018.12.1635

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Arithmetic degrees and dynamical degrees of endomorphisms on surfaces

Yohsuke Matsuzawa, Kaoru Sano and Takahiro Shibata

Vol. 12 (2018), No. 7, 1635–1657

DOI: 10.2140/ant.2018.12.1635

Abstract

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.

Keywords

arithmetic degree, dynamical degrees, arithmetic dynamics

Mathematical Subject Classification 2010

Primary: 14G05

Secondary: 11G35, 11G50, 37P05, 37P15, 37P30

Milestones

Received: 20 March 2017

Revised: 5 April 2018

Accepted: 20 June 2018

Published: 27 October 2018

Authors

Yohsuke Matsuzawa
Graduate school of Mathematical Sciences University of Tokyo Komaba, Tokyo Japan

Kaoru Sano
Department of Mathematics, Faculty of Science Kyoto University Japan

Takahiro Shibata
Department of Mathematics, Faculty of Science Kyoto University Japan

Vol. 12, No. 7, 2018