Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.