49 problems

Collatz conjecture. For every $x \in \mathbb{N}$ there exists $N \ge 0$ such that $\phi^{N}(x) = 1$. This is the classical Collatz problem, asserting that every positive integer ev…

Schwartz's conjecture. The real pentagram map might be a rare example of a Liouville–Arnold discrete integrable system.

Let $G_{\beta,n}$ be the support digraph for a discrete family, let $V_n$ be its vertex set, and let $S_n$ be the set of solution states. Write … A phase transition at…

Consider a two-dimensional subtraction ruleset with $k$ moves and regular outcomes, and let its outcome diagram decompose into segments in the sense used for two-dimensional subtra…

Let $d_n$ denote the degree growth of an integrable mapping on ${\bf C}^N$. A mapping has quadratic growth when its degree growth is of order $n^2$, and a mapping is linearizable w…

Let $(a_{k,n})$ be the normalized arrival sequence defined by the recurrence described in the source, with $a_k:=1$ as its normalization, and let $a_{k,1}$ denote its value at $1$.…

Let $\mathrm{Col}:\mathbb N\to\mathbb N$ be the Collatz map, and let $\mu$ be a finitely additive $\mathrm{Col}$-invariant measure on $\mathbb N$, with every singleton measurable.…

Maximum-period conjecture. When $\mu\bmod 3=1$, the maximum period is

Let $x$ be a positive integer. Consider the Collatz process, which sends an even integer to $x/2$ and an odd integer to $(3x+1)/2$. A computation sequence is reduced dynamics when…

Let $H_i(\mathbf{x})=\mathbf{x}^{T}A_i\mathbf{x}$ for $i=1,2$ be quadratic Nambu--Hamiltonian functions, and let…

Characteristic-sequence conjecture. For every transitive $f\in S_n$,

Discrete convex-hull conjecture. There exist $j\in\{1,2,\ldots,k\}$ and $r,s\in I_j$ such that, for some $l\leq k$,

Ordered-or-defective twin conjecture. Ordered equilibria arise only from single-string tessellations, and stable equilibria are either ordered or have defects.

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…

Akiyama–Brunotte–Pethő–Steiner conjecture. Every orbit of $r_{\varphi}$ is periodic.

Let $F_k$ be the generalized Collatz function, and call a cycle of $F_k$ non-trivial if it is not the usual trivial cycle. The source gives examples of non-trivial cycles for sever…

Let $F_k$ denote the generalized Collatz function studied in the paper, with parameters $k,n\in\mathbb{N}$. Extended Collatz conjecture. $F_k(n)$ converges for all…

Let $T_2$ be the graph appearing in the paper, and let a setting be an assignment of edge lengths satisfying the tEoM. The non-constant-solution nonexistence conjecture states that…

Let $T_q$ be the graph under consideration, let $M_q$ be a setting of its edge lengths, and let the tEoM denote the discrete time equations of motion. Assume that all edge lengths…

Stack-sorting growth conjecture. The value of this limit lies in the interval $(1.68,1.73)$.

Extremal degree conjecture. If $T\in H_0(S_n)$ is eventually constant, then

Bubble-sort degree conjecture.

Asymptotic growth conjecture. Any solution to this system grows asymptotically like

Let $q$ be an alphabet size, and let $E[D,q]$ denote the set of expansive networks on a digraph $D$ over an alphabet of size $q$, while $EA[D,q]$ denotes the set of abelian expansi…

Let $G=(V,E)$ be the graph constructed by calculating the reverse functions described in the source for all possibilities, starting from the empty graph and adding each calculated…