18 problems

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a quasi-projective variety over $k$, and let $f:X\dashrightarrow X$ be a dominant rational self-map. Let…

Let $X$ be a variety over an algebraically closed field $K$ of characteristic $p>0$, let $f$ be a rational self-map of $X$, let $x\in X(K)$ be a closed point whose orbit…

Let $f,g\in\mathbb{C}(z)\setminus\mathbb{C}$ be such that $\deg(f)\geq2$ or $\deg(g)\geq2$, and let $c\in\mathbb{C}(z)$. Define … Also let $\hat{S}_{f,g,c}^2$ denote the correspond…

Positive-characteristic dynamical Mordell–Lang conjecture. The return set $R$ is a union of finitely many infinite arithmetic progressions together with finitely many sets of the f…

Let $K$ be a field of characteristic $p$, let $X$ be a quasi-projective variety defined over $K$, let $\Phi$ be an endomorphism of $X$, let $\alpha\in X(K)$, and let $V\subseteq X$…

Dynamical Mordell–Lang conjecture. The set $S$ is a finite union of arithmetic progressions together with finitely many sets of the form

Dynamical Mordell–Lang equidistribution conjecture. If $x\in X_f(\mathbf{k})$ and $O_f(x)$ is Zariski dense in $X$, then

Let $K$ be a number field and let $X$ be a smooth projective geometrically irreducible variety over $K$. Let $f\colon X\longrightarrow X$ be a surjective morphism, let $Y\subset X$…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Let $X$ and $Y$ be quasi-projective varieties over a field $K$ of characteristic $0$. Let $f\colon X\longrightarrow Y$ be a morphism defined over $K$, let…

Let ${\mathbf{k}}$ be an algebraically closed field of characteristic $0$, let $X$ be a variety defined over ${\mathbf{k}}$, and let $f:X\dashrightarrow X$ be a rational self-map.…

Bell–Ghioca–Tucker conjecture. The set $S_{f,g}(P,Q)$ is a union of finitely many sets of the form

Let $X=\mathbb{P}^1$, and let $Y$ be a curve defined over a field $K$ of characteristic zero. Suppose that $\lambda:Y\to X$ is a finite $K$-morphism and that $\phi:X\to X$ is a mor…

Strengthened dynamical Mordell–Lang conjecture. Theorems concerning this set should hold for arbitrary quasiprojective varieties in the strengthened form in which every conclusion…

Let $\f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$, let $P\in{\mathbb P}^n({\mathbb C})$ have Zariski-dense forward orbit … and let $r\ge1$. An $(r-1)$-dimensio…

Let $X$ be a quasiprojective variety defined over ${\mathbb C}$, let $V\subset X$ be a subvariety, let $\Phi$ be an endomorphism of $X$, and let $P\in X({\mathbb C})$. The orbit of…

Let $X$ be a quasiprojective variety defined over $\mathbb{C}$, let $\Phi:X\longrightarrow X$ be any morphism, and let $\alpha\in X(\mathbb{C})$. The general dynamical Mordell–Lang…