Distributional Topological Complexity and LS-category

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