Non-density of Points of Small Arithmetic Degrees

Matsuzawa, Yohsuke; Meng, Sheng; Shibata, Toshinobu; Zhang, De‐Qi

doi:10.1007/s12220-022-01156-y

Cited by 8 publications

(1 citation statement)

References 33 publications

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“…A series of observations on the increasing geometric complexity of the iterated preimages and how, in principal, the geometric structure should exert significant influence on the underlying arithmetic structure gives rise to the expectation that the tower of

K

‐points

\begin{equation*} Y(K) \subseteq (f^{-1}(Y))(K) \subseteq (f^{-2}(Y))(K) \subseteq \cdots \end{equation*}

should eventually stabilize. This expectation has been made precise as the Preimages Question in [5, Question 8.4(1)] and solved with an affirmative answer when

X

is a smooth variety with non‐negative Kodaira dimension and

f

is étale [1, Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

Dynamical cancellation of polynomials

Zhong

2023

Bulletin of London Math Soc

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Extending the work of Bell, Matsuzawa and Satriano, we consider a finite set of polynomials over a number field and give a necessary and sufficient condition for the existence of an and a finite set such that for any , we have the cancellation result: If and are maps in such that , then in fact . Moreover, the conditions we give for this cancellation result to hold can be checked by a finite number of computations from the given set of polynomials.

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K

‐points

\begin{equation*} Y(K) \subseteq (f^{-1}(Y))(K) \subseteq (f^{-2}(Y))(K) \subseteq \cdots \end{equation*}

should eventually stabilize. This expectation has been made precise as the Preimages Question in [5, Question 8.4(1)] and solved with an affirmative answer when

X

is a smooth variety with non‐negative Kodaira dimension and

f

is étale [1, Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

Dynamical cancellation of polynomials

Zhong

2023

Bulletin of London Math Soc

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Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

Matsuzawa,

Wang

2024

Trans. Amer. Math. Soc.

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A dominant rational self-map on a projective variety is called p p -cohomologically hyperbolic if the p p -th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over Q ¯ \overline {\mathbb {Q}} , we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if f f is an 1 1 -cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a Q ¯ \overline {\mathbb {Q}} -point with generic f f -orbit is equal to the first dynamical degree of f f ; and (2) there are Q ¯ \overline {\mathbb {Q}} -points with generic f f -orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental.

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Invariant Subvarieties With Small Dynamical Degree

Matsuzawa

Meng

Shibata

et al. 2021

International Mathematics Research Notices

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Let $f:X\to X $ be a dominant self-morphism of an algebraic variety. Consider the set $\Sigma _{f^{\infty }}$ of $f$-periodic subvarieties of small dynamical degree (SDD), the subset $S_{f^{\infty }}$ of maximal elements in $\Sigma _{f^{\infty }}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty }}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with SDD and give an optimal upper bound $$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$as $n\to \infty $, where $d_1(f)$ is the 1st dynamic degree. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound $$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$as $n \to \infty $, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.

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Non-density of Points of Small Arithmetic Degrees

Cited by 8 publications

References 33 publications

Dynamical cancellation of polynomials

Dynamical cancellation of polynomials

Arithmetic degrees and Zariski dense orbits of cohomologically hyperbolic maps

Invariant Subvarieties With Small Dynamical Degree

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