Arithmetic degrees and dynamical degrees of endomorphisms on surfaces

2017/01/16 by Matsuzawa, Yohsuke, Sano, Kaoru, Shibata, Takahiro · 1 citation
#11G35 #11G50 #14G05 #14G25 #14G40 #37P55 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1701.04369

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, we prove that there exists a Zariski dense set of rational points having disjoint orbits if the endomorphism is an automorphism.

Cited by

• Rational self-maps of projective surfaces with a regular iterate

Related

• Higher arithmetic degrees of dominant rational self-maps
• Arithmetic degree and its application to Zariski dense orbit conjecture
• Examples of dynamical degree equals arithmetic degree
• Zariski density of points with maximal arithmetic degree
• Arithmetic degrees for dynamical systems over function fields of characteristic zero