# Canonical height functions for affine plane automorphisms

@article{Kawaguchi2006CanonicalHF,
title={Canonical height functions for affine plane automorphisms},
author={Shu Kawaguchi},
journal={Mathematische Annalen},
year={2006},
volume={335},
pages={285-310}
}
• Shu Kawaguchi
• Published 7 April 2006
• Mathematics
• Mathematische Annalen
Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.

### Canonical heights for plane polynomial maps of small topological degree

• Mathematics
• 2012
We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the

### The Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane

• Mathematics
• 2020
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of

### Local and global canonical height functions for affine space regular automorphisms

Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic

### Ample canonical heights for endomorphisms on projective varieties

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by

### Canonical Height Functions For Monomial Maps

• Mathematics
• 2012
We show that the canonical height function defined by Silverman does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain

### An upper bound for the height for regular affine automorphisms of A^n

In 2006, Kawaguchi proved a lower bound for height of h(f(P)) when f is a regular affine automorphism of A^2, and he conjectured that a similar estimate is also true for regular affine automorphisms

### Periodic points and arithmetic degrees of certain rational self-maps

. Consider a cohomologically hyperbolic birational self-map deﬁned over the algebraic numbers, for example, a birational self-map in dimension two with the ﬁrst dynamical degree greater than one, or

### Regular polynomial automorphisms in the space of planar quadratic rational maps

• Mathematics
• 2020
In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms $${\mathcal {H}}_2^2$$ in the semistable locus of the moduli space of quadratic rational maps, using

### On the complex dynamics of birational surface maps defined over number fields

• Mathematics
• 2015
We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford-Diller energy

## References

SHOWING 1-10 OF 12 REFERENCES

### On the degree of iterates of automorphisms¶of the affine plane

Abstract:For a polynomial automorphism f of ?2ℂ, we set τ = deg f2)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number

### Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles

Abstract We construct canonical heights of subvarieties for dynamical eigensystems of several morphisms for line bundles over a number field, and study some of their properties. We also construct

### Rational points on K3 surfaces: A new canonical height

A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the

### Dynamical properties of plane polynomial automorphisms

• Mathematics
Ergodic Theory and Dynamical Systems
• 1989
Abstract This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

### Geometric and arithmetic properties of the Hénon map

The real and complex dynamics of the H6non map 4) : Ik2--*~ 2, 4 ) ( x , y ) = ( y , y 2 + b + a x ) (1) have been extensively studied since H6non [6] showed that such maps may have strange

### A compactification of Hénon mappings inC2 as dynamical systems

• Mathematics
• 1997
In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball such that the H\'enon mapping extends continuously. That paper used a delicate analysis of some

### Canonical heights on varieties with morphisms

• Mathematics
• 1993
© Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions

### Sur la dynamique arithmétique des automorphismes de l’espace affine

Nous etudions les proprietes arithmetiques des iteres de certains automorphismes polynomiaux affines. Nous traitons des questions concernant les points periodiques et non-periodiques, en particulier