Vol. 12, No. 7, 2018

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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

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.

arithmetic degree, dynamical degrees, arithmetic dynamics
Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11G35, 11G50, 37P05, 37P15, 37P30
Received: 20 March 2017
Revised: 5 April 2018
Accepted: 20 June 2018
Published: 27 October 2018
Yohsuke Matsuzawa
Graduate school of Mathematical Sciences
University of Tokyo
Komaba, Tokyo
Kaoru Sano
Department of Mathematics, Faculty of Science
Kyoto University
Takahiro Shibata
Department of Mathematics, Faculty of Science
Kyoto University