14 problems

Let ${\cal{A}}=\{A_1,\ldots,A_m\}$ be a family of $m$ real $d\times d$ matrices, and let ${\cal{Q}}_m$ denote the space of all such families, identified with ${\mathbb R}^{m d^2}$.…

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

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

A binary matrix is a matrix whose entries belong to $\{0,1\}$. For a finite matrix family, the finiteness property means that some finite product attains the joint spectral radius.…

Let $\mathcal F$ be any finite set of square matrices with rational entries, and let $\overline\rho_k(\mathcal F)$ denote the maximum normalized spectral radius among products of l…

Let the ambient space consist of finite families of complex square matrices, equipped with its natural measure, and let the finiteness property mean that some finite product attain…

Let $\mathcal B$ be the space of bounded families of matrices, $\mathcal R$ the subset of reducible families, and $\mathcal D\subsetneq\mathcal R$ the subset of defective families.…

Maximal-growth trajectory conjecture. For a generic discrete-time system, there exists a trajectory $x(k)$ such that

Let $\mathbb{A}\in\{0,1\}^{K\times K}$ be any matrix, and let … with its natural subspace topology. Here spectral finiteness restricted to $\mathbb{A}$ means that the joint spectra…

Let $T\colon W\rightarrow W$ have the closing by periodic orbits property, and let $\boldsymbol{C}\in\operatorname{C}(W,\mathbb{R}^{d\times d})$ be a continuous matrix-valued funct…

Let $\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times d}$, with $d\geq 4$, be an arbitrary pair satisfying condition (1.3a), namely that the matrices share a common non-strict…

Let $r$ be an odd integer, and let $A_1,A_2$ be the $(r-1)\times(r-1)$ matrices defined by … and $(A_s)_{ij}=0$ otherwise, for $s=1,2$. A product is an s.m.p. if it is a spectrum-m…

Let $A_1,A_2$ be the special $20\times20$ matrices with nonnegative integer entries associated with overlap-free words, and let $\mathcal{M}=\{A_1,A_2\}$. Let $\beta$ be the upper…

Let $r,d \in \mathbb{N}$, let $\mathcal{I}_r(\mathbb{K}^d)$ be the ambient class of $r$-tuples under consideration, and let $\Omega_r^n$ denote the words of length $n$. For…