26 problems

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

Bellon–Viallet conjecture. The entropy $epsilon$ is the logarithm of an algebraic integer for every birational self-map of a projective space.

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…

Algebraic-integrality conjecture. The entropy is always the logarithm of an algebraic integer, both for maps and for the lattice equations considered here.

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

Liouville integrability conjecture. The deformed type $D_6$ map is a Liouville integrable map.

Integrability conjecture. For each $N$, the deformed type $D_{2N}$ map is integrable.

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…

For a discrete dynamical system with the Laurent property, let the logarithmic Mahler measure mean the logarithmic Mahler measure of its iterates, and let the algebraic entropy be…

Nilpotent-group Addition Theorem conjecture. $AT(G)$ holds for every nilpotent group $G$.

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…

Let $G$ be an amenable group, $A$ an abelian group, and let $G$ act on $A$ on the left by $\alpha$. Let $H$ be a non-trivial subgroup of $G$ of infinite index. Infinite-index subgr…

Let $G$ be an amenable group, $A$ an abelian group, and let $G$ act on $A$ on the left by $\alpha$. Let $H$ be a subgroup of $G$. Restriction conjecture. One has … This is proposed…

Algebraic-entropy criterion. The algebraic entropy of $$ should be positive if and only if

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…

Growth-rate conjecture. If $Q$ admits a strictly subadditive labeling, then $f$ is bounded and periodic; otherwise, if $Q$ admits a subadditive labeling, then $f$ grows exponential…

Eigenvalue read-off conjecture. If the deautonomisation is sufficiently general, then the value of the largest eigenvalue of the linear map on the Picard group, and hence the algeb…

For odd $k=3,5,7,\ldots$, consider the non-autonomous mapping … The mapping passes the singularity confinement test. Full-deautonomisation entropy conjecture. Its algebraic entropy…

Consider the extended Hietarinta–Viallet mapping … and let $\lambda_k$ denote its algebraic entropy. For $k=2$, this is the original Hietarinta–Viallet equation. Hietarinta–Viallet…

For a rational map, let $d_n$ denote the degree of its $n$th iterate and define its algebraic entropy by … A map is Liouville–Arnold integrable when it has the appropriate number o…

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…