Related papers: Kawaguchi-Silverman conjecture for int-amplified e…

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective…

Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $\tau: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every…

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$…

We first show that the moniod of separable surjective self morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi…

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q >…

Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model…

Let $X$ be a variety defined over an algebraically closed field $k$ of characteristic $0$, let $N\in\mathbb{N}$, let $g:X\dashrightarrow X$ be a dominant rational self-map, and let $A:\mathbb{A}^N\to \mathbb{A}^N$ be a linear transformation…

We prove that any surjective self-morphism with $\delta_f > 1$ on a potentially dense smooth projective surface defined over a number field $K$ has densely many $L$-rational points for a finite extension $L/K$.

An endomorphism $f$ of a projective variety X is polarized (resp. quasi-polarized) if $f^*H$ is linearly equivalent to $qH$ for some ample (resp. nef and big) Cartier divisor $H$ and integer $q > 1$. First, we use cone analysis to show that…

Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to{X}$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point…

For endomorphisms of projective varieties, we prove that the arithmetic degree of a point with Zariski dense orbit must be a cohomological Lyapunov multiplier of the dynamical system. We will apply our result to deduce a corollary towards…

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short),…

Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic…

In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then…

Let $A$ be an abelian variety defined over $\bar{\mathbb{Q}}$, and let $\varphi$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $\varphi$, or there…

Let $X$ be a smooth projective variety over an algebraically closed field, and $f\colon X\to X$ a surjective self-morphism of $X$. The $i$-th cohomological dynamical degree $\chi_i(f)$ is defined as the spectral radius of the pullback…

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi-Yau type, i.e., $(X,\Delta)$ is…

We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height…

We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on P^N so that (S,D)-integral points in each orbit are…

Let $X$ be a compact K\"ahler manifold and let $f:X\rightarrow X$ be a dominant rational map which is 1-stable. Let $\lambda_1$ and $\lambda_2$ be the first and second dynamical degrees of $f$. If $\lambda_1^2>\lambda_2$, then we show that…