25 problems

For a discrete equation, a singularity is confined when any spontaneously appearing singularity disappears again after a finite number of iteration steps. Singularity confinement c…

Let $0<\alpha=\beta<\frac{1}{2}$ and set $\gamma=1-2\alpha$, so that $0<\gamma<1$. Consider the hexagonal circle patterns $z^{3\alpha}$ with an intersection point at the origin and…

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

Let the Lyness system be an $N$th-order mapping, and interpret integrability in the sense made precise in the paper. Liouville integrability conjecture. The Lyness map is Liouville…

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 a collimation chamber be a region in the stability-condition space of the BPS system, and consider the TBA equation for the functions $\mathcal{X}_\gamma(\epsilon)$ and its cor…

Let $m\in\mathbb{Z}$ and let $\tilde z$ be an $m$-closed polygon recutting lattice map, meaning $\tilde z_{i,j}=\tilde z_{i+m,j-m}$ for all $i,j$. Assume $\tilde z_{i,0}=0$ for all…

Let $\mathcal{L}$ be the lattice on which the dSKP recurrence is defined, let $x:\mathcal{L}\to\hat{\mathbb{C}}$ be a solution, and set $a_{i,j}=x(i,j,[i+j]_2)$. Suppose…

Let $m\geq 1$, let $\tilde{f}:\mathbb{Z}^2\rightarrow {\mathbb{R}\mathrm{P}}^2$ be a $2m$-closed pentagram lattice map, equivalently let…

Consider the generalized Jacobi spectral problem … Let $u_n^\delta$ and $w_n$ have the power-series expansion in $\alpha$ … … where $k$ is an arbitrary nonzero constant and $I$ is…

Let $T_{I,J}$ be a generalized pentagram map associated with two multi-indices $I$ and $J$. A map is Hamiltonian here if it preserves a certain nontrivial Poisson structure. Khesin…

Let $(v_i)$ be a twisted polygon in $b2b2\text{?}b2b2$ and let $I$ be a multi-index defining a generalized pentagram map $T_{I,J}$. The map $T_{I,I}$ is the diagonal case in which…

Consider the same hypergeometric weights on the shifted lattice $\mathbb{N}+1-\gamma$, and let $(\hat{x}_n,\hat{y}_n)$ be the corresponding recurrence variables. Shifted-lattice as…

Asymptotic conjecture. For this initial value,

A biperiodic circle pattern is a square-grid circle pattern invariant under two independent lattice periods. Miquel integrability conjecture. For biperiodic circle patterns, Miquel…

For the extended Hietarinta–Viallet mapping … consider even $k=4,6,8,\ldots$ and resolve the mapping by blowing up points of $\mathbb{P}^1\times\mathbb{P}^1$ until it becomes birat…

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…

The source discusses irreducibility and co-primeness of iterated terms for discrete mappings, and relates these properties to singularity structure and integrability. In particular…

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…

Consider a non-autonomous mapping regularised by blowing up its indeterminacy points, and the induced map on the exceptional curves. The singularity-confinement conditions are obta…

Let $\ell$ and $\ell'$ be the non-negative integers specifying the blow-up patterns, and let the matrix governing the evolution of the mapping's parameters be compared with the mat…

Let $\tau(n_1,n_2,\ldots)$ be a tau function of the fully discrete KP hierarchy. For an index $i$, write $\tau_{(\pm i)}$ for the shift of $\tau$ by $\pm1$ in the variable $n_i$, a…

The paper considers discrete models generated from simple algebraic identities, nonlinear superpositions of the Riccati equation, homogeneous decompositions of the angle bisector t…