Nonlinear biseparating maps

An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains $\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of biseparating given above turns out to be too weak to determine the structure of $T$. In this paper, we propose a revised definition of biseparating maps for general nonlinear operators acting between spaces of vector-valued functions, which coincides with the previous definition for additive maps. Under some mild assumptions on the function spaces involved, it turns out that a map is biseparating if and only if it is locally determined. We then delve deeply into some specific function spaces -- spaces of continuous functions, uniformly continuous functions and Lipschitz functions -- and characterize the biseparating maps acting on them. As a by-product, certain forms of automatic continuity are obtained. We also prove some finer properties of biseparating maps in the cases of uniformly continuous and Lipschitz functions.