12 problems

Let $f$ be a rational map, and let its algebraic entropy be the exponential growth rate of the degrees of its iterates. Bellon–Viallet conjecture. The algebraic entropy of every ra…

Integrability conjecture. The discrete dynamics induced by the iteration of cluster maps $\psi_{D_{2N}}$ is a discrete integrable system.

Let ${\cal A}$ be a cluster algebra, or let $\varphi$ be a map with the Laurent property obtained from a cluster algebra with periodicity in the sense of the cited constructions. L…

Locally finite-group Addition Theorem conjecture. If $G$ is a locally finite group, then $AT(G)$ holds.

Let $S$ be a cancellative left amenable monoid acting on the compact abelian group $K$ on the left by $\gamma$. Let $\widehat\gamma$ denote the induced dual action on the Pontryagi…

Let $S$ be a cancellative right amenable monoid acting on the abelian group $A$ on the left by $\alpha$. Let $B$ be an $\alpha$-invariant subgroup of $A$, and write $\alpha_B$ and…

A birational map is assumed to admit an algebraic entropy, measuring the asymptotic growth of the degrees of its iterates. Algebraic entropy conjecture. The condition that algebrai…

Octagon-poset tau-sequence conjecture. For each octagon poset $P$, the $R$-system associated with $G(P)$ admits a strong $\tau$-sequence consisting of irreducible Laurent polynomia…

Let $G$ be a finitely generated group such that $h_{alg}(id_G)=\infty$, equivalently, $G$ has exponential growth. Infinite algebraic entropy conjecture. Then $h_{alg}(\phi)=\infty$…

Spectral-radius refinement conjecture. There are two possibilities: (1) if $|\lambda_{\max}|=\rho({\bf M}_\pm)>\rho({\bf M}_\mp)\geq1$, then $k=1$ and the symbol is repeated,…

Let $(d_n)$ be a sequence satisfying … with initial conditions $d_1=-1$ and $d_2=\cdots=d_N=0$ up to shifting the index, and suppose that $(d_n)$ is not periodic. Let…

Let $b_{1,j+1}$ be the coefficients defining a birational map … , and let $[x]_+=\max(x,0)$. Define … Let $\lambda_{\max}$ be the root of largest magnitude among the roots of these…