Distributional topological complexity and LS-category

Abstract

We define a new version of topological complexity $(\mathsf{TC})$ of a space, denoted as $\mathsf{dTC}$, which we think fits better for motion planning for some autonomous systems. Like topological complexity, $\mathsf{dTC}$ is also a homotopy invariant. Also, $\mathsf{dTC}$ has a corresponding analogue, denoted as $\mathsf{dcat}$, to the Lusternik–Schnirelmann category $(\mathsf{cat})$. In this paper, we do computations and estimates for both $\mathsf{dTC}(X)$ and $\mathsf{dcat}(X)$ for some spaces $X$ as well as a comparison with $\mathsf{TC}(X)$ and $\mathsf{cat}(X)$.