If $ (f_n)_{n \in \mathbb N}$ are smooth maps $ f_n \colon \mathbb R^d \to \mathbb R^d$ with Jacobian matrix $ \nabla f_n$ equi-bounded (say in in uniform norm) then by Arzela-Ascoli we have that, up to subsequences, they converge locally to some continuous function $ f_n \colon \mathbb R^d \to \mathbb R^d$ .

What happens is we weaken the assumption by only requiring that some minors of the matrix $ \nabla f_n$ are uniformly bounded? And what if we only have $ \sup_n |\det \nabla f_n | < \infty$ ?

Instead of using Arzela-Ascoli one could rely on compactness of functions of bounded variation in $ L^1$ , but this would in any case require a $ L^1$ -bound (not $ L^\infty$ ) on the *full* Jacobi matrix $ \nabla X_n$ .