31 problems

Let $\mathbb{A}=(A,B)\in\mathbf{G}^2$, where $\mathbf{G}=\{X\in\mathrm{GL}(2,\mathbb{R}):\det X>0\}$, and let $\hat{\rho}(\mathbb{A})$ be the joint spectral radius, defined by … A…

Bochi–Fayad conjecture. The set of all resistant pairs $(H,R)\in\mathcal{H}\times\mathcal{R}$ has full Lebesgue measure. Partial results are known, and explicit resistant pairs can…

Binary-matrix finiteness conjecture. Every pair of binary matrices has the finiteness property.

Let a set of matrices have the finiteness property when its joint spectral radius is attained by some finite product. Finiteness conjecture. It was conjectured that all sets of mat…

Let $A_0$ and $A_1$ be the matrices defining the joint spectral radius $\rho$, and let $\Pi_n=A_{i_1}\cdots A_{i_n}$ be a product of length $n$. Finiteness conjecture. Numerical ob…

Let $S=\{v_i\}_{i=1}^g\subseteq\matrices$ be a collection of matrices, and define its Wielength by … where $S^k=\{v_\sigma:\sigma\in\{1,\dots,g\}^k\}$,…

Let $(Ai){i e i}$ be a completely reducible family in $operatorname{GL}d(mathbb{R})^{mathcal{I}}$, with $kin{0,ldots,d-1}$ and $sin(k,k+1)$. Let $p$ be the number of functions in t…

Locally uniform trajectory-bounds conjecture. There is a constant $\Theta_d>0$, depending only on $d$ and the norm, such that for every $\mathcal{M}\in\mathcal{H}(d)$ with…

Product conjecture. The product of the $u_i$-tilted matrices

Maesumi's conjecture. Lebesgue-almost every finite family of matrices has an SMP. The conjecture was proposed after the finiteness conjecture was refuted by Bousch and Mairesse; th…

Undecidability conjecture. It is undecidable to check whether $\rho\geq 1$ for the joint spectral radius $\rho$.

Let $\mathcal{A}=(A_1,\ldots,A_d)$ be a $d$-tuple of $D \times D$-matrices, and assume that there is an $N$ such that the linear span of…

Let $n$ be a positive integer and let $\mathcal{S}$ be an SIA set of $n \times n$ stochastic matrices. Its SIA-index is the length of the shortest SIA product formed from matrices…

Finite-variance no-outlier conjecture. The spectral radius of $P_n$ is bounded above by $\sigma+o(1)$ in probability as $n\to\infty$, assuming only that the atom variables…

Moore's conjecture. If an equilibrium state associated to an irreducible tuple of matrices is ergodic but not totally ergodic with respect to $\sigma$, then it must fail to be ergo…

Local overline{d}-continuity conjecture. The map

Mixing conjecture. If for some $s>0$ the equilibrium state of $(\mathsf{A},s)$ is ergodic with respect to $\sigma^d$, then it is mixing with respect to $\sigma$. If there exists…

Algorithmic unsolvability conjecture. The recognition problem for CP families with rational matrix entries is algorithmically unsolvable.

Let $M_2(\mathbb{R})$ be the space of all $2\times2$ real matrices, and let … be the lower spectral radius as a function on pairs of matrices. Discontinuity-dimension conjecture. T…

Fix real parameters $p\in(0,1)$ and $q>1$, and write $\mathfrak{r}_q(A_1,A_2,p)$ for the generalized Rényi dimension associated with the probability vector $(p,1-p)$. Discontinuity…

Higher-dimensional singular value pressure conjecture. For each $d\geq 1$, there exist an integer $n(d)$ and a continuous function $\hat{K}_d\colon(0,+\infty)\to(0,+\infty)$ such t…

Blondel et al.'s conjecture. The set of parameters $\alpha\in[0,1]$ for which $\mathcal F_\alpha$ does not satisfy the finiteness property has measure zero. The conjecture is prese…

Positive-measure discontinuity conjecture. The set of discontinuities of $\check{\varrho}$ has positive Lebesgue measure. This is presented as a weaker version of the resist-impuri…

The resist-impurities conjecture. The set of pairs $(H,R)\in\mathcal{H}\times\mathcal{E}$ that resist impurities has full Lebesgue measure in $\mathcal{H}\times\mathcal{E}$. This c…

Let $n,d\geq 1$, let $\mathsf{A}\in\mathit{GL}_d(\mathbb{R})^n$, and let $\ell(\mathsf{A})$ be the smallest index of domination for $\mathsf{A}$. The continuity criterion conjectur…