48 problems

Let $X$ be a projective variety over $k$, and let $\phi:X\rightarrow X$ be a polarized endomorphism. The forward orbit of $x\in X(k)$ is $\{\phi^m(x):m\geq 0\}$. Zhang's Zariski de…

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 $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map, and let … be its dynamical degree. Bellon–Viallet conjecture. The dynamical degree is an algebraic integ…

Let $d\geq 2$ and $n\geq 2$. A real morphism $f:\mathbb{P}^n\rightarrow\mathbb{P}^n$ is a morphism defined over $\mathbb{R}$, where $\mathbb{P}^n$ is complex projective $n$-space.…

Ghioca–Tucker–Zhang's conjecture. $Y$ is preperiodic under $f$ if and only if there exists a Zariski-dense subset of smooth preperiodic points $x \in Y$ such that the tangent subsp…

Let $E$ be a rational elliptic curve, let $Q\in E(\mathbb{Q})$ be a rational point with $x(Q)=a/b$, and let $\psi_n$ denote the $n$-th division polynomial. A compact $\mathbb{Z}^2$…

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

Dynamical realization conjecture. There exists a variety $X$ defined over $\mathbb Q$ and a map $F:X\to X$ such that there is a family of bijections

Shape-detection conjecture. If all mixing shapes of cardinality $r$ for $\alpha$ are mixing, then $\alpha$ is mixing of order $r$: whenever…

Schmidt's conjecture. If $h(\alpha)=0$, then $\phi$ is $\lambda_X$-almost everywhere equal to an affine map. In particular, measurable conjugacy implies algebraic conjugacy.

A dynamical degree is the birational invariant associated with the growth of the iterates of a self-map of an algebraic variety. In the cases discussed in the source, dynamical deg…

A klt Fano variety is a variety with Kawamata log terminal singularities whose anticanonical divisor is ample. A finite quotient of a toric variety is a variety obtained as the quo…

A surjective endomorphism of a projective algebraic variety $X$ is a morphism $f\colon X\to X$. It is int-amplified if there exists an ample Cartier divisor $A$ on $X$ such that…

Let $(X,c6)$ be a non-autonomous algebraic dynamical system, and suppose that some power $(X^n,c6^{\times n})$ has a non-diagonal proper irreducible invariant subvariety. Four-dime…

Let $k$ be a field, let $n$ be a positive integer, and let $f\in\operatorname{End}(\mathbb{A}^n_k)$. Its dynamical degree is … where $\deg(f)$ is the maximum of the degrees of the…

Let $T$ be an irreducible variety, let $X/T$ be a family of smooth irreducible projective varieties, and let $f:X/T\dashrightarrow X/T$ be a family of dominant rational maps. Write…

Generic billiard dynamical-degree conjecture. The dynamical degree of the generic billiard satisfies

Let $f$ be an endomorphism of a smooth variety $X$ over a field ${\mathbf{k}}$ of positive characteristic, and let $\lambda_i(f)$ denote its $i$-th dynamical degree. Let…

Let $X$ be a normal projective variety and let $f:X\to X$ be a non-isomorphic surjective endomorphism. The properties weakly $\delta$-imprimitive, strongly imprimitive, quasi-abeli…

A projective variety is int-amplified if it admits an endomorphism $f\colon X\to X$ and an ample divisor $H$ such that $f^*H-H$ is ample. Fakhruddin–Meng–Zhang–Zhong's conjecture.…

Ih–Rees–Silverman conjecture. When $k\geq2$, the PCF endomorphisms are not Zariski dense in the space $\operatorname{End}_d^k$ of all degree-$d$ endomorphisms of $\mathbb{P}^k$.

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic zero, and let $f:X\dasharrow X$ be a dominant rational self-map. Write $k(X)^f$ for the fie…

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a dominant rational self-map. Assume that…

Let $X$ be a smooth projective variety defined over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a birational self-map. Its first d…

Positive-characteristic Zariski dense orbit conjecture. At least one of the following holds: