8 problems

Let $\mathcal{S}$ be the nontrivial symmetric invariant set of the map $G_{\varepsilon,3}$, and let $\varepsilon$ denote the coupling parameter. Invariant-sets-inside-$\mathcal{S}$…

Symmetric invariant-set conjecture. At every $\varepsilon_n$, new symmetric invariant sets appear; these sets are contained in $\mathcal{S}$ and are symmetric. The conjecture is mo…

Second asymmetric-set conjecture. There exists such a polyhedron $P_3$ for which $\mathcal{A}_2$ is symmetric with respect to $S_1,S_2$ and $S_4$, asymmetric with respect to…

Let $G_{\varepsilon,3}:\mathbb{T}^3\to\mathbb{T}^3$ be the piecewise affine map of the four-site globally coupled system, with coupling parameter $\varepsilon$, and let $S_G$ denot…

Let $G_{\varepsilon,3}:\mathbb{T}^3\to\mathbb{T}^3$ be the piecewise affine map describing the four-site system, and let $\varepsilon^*$ be the critical coupling value at which the…

Connectedness conjecture. For any of the three models described, and for any equi-parameter $c\in\mathbb{C}$, the uni-J set is connected only if $c$ is in the equi-M set of the net…

Ergodicity and component-count conjecture. For all $N\in\mathbb{N}$, the system $(F_{\varepsilon,N},\mathbb{T}^N)$ is ergodic if $\varepsilon<\varepsilon_0$, for some…

Ergodicity conjecture for three sites. There is a unique ergodic absolutely continuous invariant measure for $\varepsilon<\frac{4-\sqrt{10}}{2}$, and the number of ergodic componen…