Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties

Matsuzawa, Yohsuke; Sano, Kaoru

doi:10.1017/etds.2018.117

Cited by 10 publications

(6 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2 for semiabelian varieties is proved in Theorem 1.5 and KSC 1.3 for them is proved in [MS20]. So AZO 1.2 and KSC 1.3 hold for (X, f ) too (cf.…”

Section: Endomorphisms Descending Along a Fibrationmentioning

confidence: 86%

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

Jia¹,

Shibata²,

Xie³

et al. 2021

Preprint

View full text Add to dashboard Cite

Let X be a quasi-projective variety and f : X → X a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree d 1 (f ) of f equals the arithmetic degree α f (P ) at a point P having Zariski dense f -forward orbit. Assuming X is a smooth affine surface, such that the log Kodaira dimension κ(X) is non-negative (resp. the étale fundamental group π ét 1 (X) is infinite), we confirm AZO, (hence) ZDO, and KSC (when deg(f ) ≥ 2) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with "larger" first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).

show abstract

“…2 for semiabelian varieties is proved in Theorem 1.5 and KSC 1.3 for them is proved in [MS20]. So AZO 1.2 and KSC 1.3 hold for (X, f ) too (cf.…”

Section: Endomorphisms Descending Along a Fibrationmentioning

confidence: 86%

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

Jia¹,

Shibata²,

Xie³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Conjecture 2.7 is verified in several cases (not only for endomorphisms, but also for dominant rational maps). See [24,Remark 1.8], [14,19,20,22,23,29].…”

Section: Arithmetic Degree and Kawaguchi-silverman Conjecturesmentioning

confidence: 99%

Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

Matsuzawa¹,

Yoshikawa²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proof of this fact is essentially covered in [MS20]. First of all, the dynamical degree of equals the dynamical degree of (since each iterate of is a composition of with a suitable translation).…”

Section: Introductionmentioning

confidence: 99%

“…First of all, the dynamical degree of equals the dynamical degree of (since each iterate of is a composition of with a suitable translation). Second, if and only if the spectral radius of is equal to , and so all roots of the polynomial P must have absolute value equal to (for more details, see [MS20]). Then a classical theorem of Kronecker regarding algebraic numbers whose Galois conjugates all have absolute value equal to yields that all roots of must be roots of unity, as desired.…”

Section: Introductionmentioning

confidence: 99%

A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

Bell¹,

Ghioca

2022

Can. Math. Bull.

View full text Add to dashboard Cite

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety X defined over an algebraically closed field K of characteristic 0, endowed with a birational self-map φ of dynamical degree 1, we expect that either there exists a non-constant rational function f : X P 1 such that f • φ = f , or there exists a proper subvariety Y ⊂ X with the property that for any invariant proper subvariety Z ⊂ X, we have that Z ⊆ Y . We prove our conjecture for automorphisms φ of dynamical degree 1 of semiabelian varieties X. Also, we prove a related result for regular dominant self-maps φ of semiabelian varieties X: assuming φ does not preserve a non-constant rational function, we have that the dynamical degree of φ is larger than 1 if and only if the union of all φ-invariant proper subvarieties of X is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.

show abstract

Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties

Abstract: We prove a conjecture by Kawaguchi-Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties. Contents

Cited by 10 publications

References 19 publications

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

Contact Info

Product

Resources

About