Higher arithmetic degrees of dominant rational self-maps

Suppose that f : X --> X is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if PX(Q) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(f) of f if the orbit of P is Zariski dense in X.

In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of X. We begin by defining a set of arithmetic degrees of f, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

Higher arithmetic degrees of dominant rational self-maps, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), Vol. XXIII (2022), 463-481, https://doi.org/10.2422/2036-2145.201908_014.

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Work Title Higher arithmetic degrees of dominant rational self-maps
Access
Open Access
Creators
  1. Nguyen-Bac Dang
  2. Dragos Ghioca
  3. Fei Hu
  4. John Lesieutre
  5. Matthew Satriano
Keyword
  1. Dynamical degree
  2. Arithmetic degree
  3. Arithmetic varieties
  4. Heights
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Scuola Normale Superiore - Edizioni della Normale
Publication Date March 30, 2022
Publisher Identifier (DOI)
  1. 10.2422/2036-2145.201908_014
Source
  1. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
Deposited August 26, 2022

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Version 1
published

  • Created
  • Added higherks-1.pdf
  • Added Creator Nguyen-Bac Dang
  • Added Creator Dragos Ghioca
  • Added Creator Fei Hu
  • Added Creator John Lesieutre
  • Added Creator Matthew Satriano
  • Published
  • Updated Description Show Changes
    Description
    • Suppose that \(f \colon X atmap X\) is a dominant rational self-map of a smooth projective variety defined over \({\overline{\mathbf Q}}\). Kawaguchi and Silverman conjectured that if \(P \in X({\overline{\mathbf Q}})\) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $\lambda_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of \(X\). We begin by defining a set of arithmetic degrees of \(f\), independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
    • Suppose that $$\(f \colon X atmap X\)$$ is a dominant rational self-map of a smooth projective variety defined over $$\({\overline{\mathbf Q}}\)$$. Kawaguchi and Silverman conjectured that if $$\(P \in X({\overline{\mathbf Q}})\)$$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $$\lambda_1(f)$$ of $$f$$ if the orbit of $$P$$ is Zariski dense in $$X$$.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $$\(X\)$$. We begin by defining a set of arithmetic degrees of $$\(f\)$$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated Description Show Changes
    Description
    • Suppose that $$\(f \colon X atmap X\)$$ is a dominant rational self-map of a smooth projective variety defined over $$\({\overline{\mathbf Q}}\)$$. Kawaguchi and Silverman conjectured that if $$\(P \in X({\overline{\mathbf Q}})\)$$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $$\lambda_1(f)$$ of $$f$$ if the orbit of $$P$$ is Zariski dense in $$X$$.
    • Suppose that \\( \(f \colon X atmap X\) \\) is a dominant rational self-map of a smooth projective variety defined over $$\({\overline{\mathbf Q}}\)$$. Kawaguchi and Silverman conjectured that if $$\(P \in X({\overline{\mathbf Q}})\)$$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $$\lambda_1(f)$$ of $$f$$ if the orbit of $$P$$ is Zariski dense in $$X$$.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $$\(X\)$$. We begin by defining a set of arithmetic degrees of $$\(f\)$$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated Description Show Changes
    Description
    • Suppose that \\( \(f \colon X atmap X\) \\) is a dominant rational self-map of a smooth projective variety defined over $$\({\overline{\mathbf Q}}\)$$. Kawaguchi and Silverman conjectured that if $$\(P \in X({\overline{\mathbf Q}})\)$$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $$\lambda_1(f)$$ of $$f$$ if the orbit of $$P$$ is Zariski dense in $$X$$.
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* \in *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $\lambda_1$(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $$\(X\)$$. We begin by defining a set of arithmetic degrees of $$\(f\)$$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated Description Show Changes
    Description
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* \in *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $\lambda_1$(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* \in *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated Description Show Changes
    Description
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* \in *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated

Version 2
published

  • Created
  • Updated Description Show Changes
    Description
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* ∈ *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • Suppose that $f : X \dashrightarrow X$ *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* ∈ *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Published
  • Updated

Version 3
published

  • Created
  • Updated Description Show Changes
    Description
    • Suppose that $f : X \dashrightarrow X$ *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* ∈ *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • Suppose that *f* : *X* atmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* ∈ *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Published
  • Updated Work Title, Source, Keyword, and 1 more Show Changes
    Work Title
    • Higher arithmetic degrees of rational maps
    • Higher arithmetic degrees of dominant rational self-maps
    Source
    • ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
    • Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
    Keyword
    • Dynamical degree, Arithmetic degree, Arithmetic varieties, Heights
    Description
    • Suppose that *f* : *X* \ratmap *X* is a dominant rational self-map of a smooth projective variety defined over {\overline{\mathbf Q}}. Kawaguchi and Silverman conjectured that if *P* ∈ *X*({\overline{\mathbf Q}}) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • Suppose that *f* : *X* --> *X* is a dominant rational self-map of a smooth projective variety defined over Q. Kawaguchi and Silverman conjectured that if *P* ∈ *X*(Q) is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree λ_1(*f*) of *f* if the orbit of *P* is Zariski dense in *X*.
    • In this note, we extend the Kawaguchi--Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of *X*. We begin by defining a set of arithmetic degrees of *f*, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.
  • Updated