Invertibility, Often
arXiv:2603.26376
The paper proves that in four different dynamical‑system settings—the space of surjective continuous maps on Cantor sets, measure‑preserving maps on Polish spaces, L¹ representations of nonsingular maps, and continuous measure‑preserving maps on Cantor spaces with good measures—the set of invertible (isomorphic) maps forms a residual (comeager) subset of all maps.
Abstract
By using a similar pattern of arguments, we show that in four categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at measure preserving maps on Polish measure spaces. We then consider the representations of nonsingular maps on Polish measure spaces. Finally, we examine continuous, measure preserving maps on Cantor spaces equipped with so-called good measures.