The n-dimensional classification conjecture for bounded omega-limit sets

Let $T$colon $\mathbb{R}^n\to\mathbb{R}^n$ be an nD map as given in Definition 1, and let $\mathcal{A}$ be a bounded $\omega$-limit set different from the fixed point $O$ or from related local invariant sets when $O$ is nonhyperbolic. A bounded omega-limit classification conjecture. Such a set can only be one of the following, which may coexist: a nonhyperbolic $k$-cycle with $k\geq 2$, occurring in $m$-dimensional sets with $m<n$ filled with cycles of the same symbolic sequence; a finite number of $m$-dimensional sets with $m<n$ filled with quasiperiodic orbits; or an $m$D weird quasiperiodic attractor, where $2\leq m\leq n$. When no cycles exist, namely in the latter two cases, $\mathcal{A}$ exhibits weak sensitivity to initial conditions. This conjecture proposes a general classification of bounded attractors in the authors' class of piecewise linear discontinuous maps, extending the observed two-dimensional behavior to higher dimensions; its status is not resolved in the supplied text.