# Sequence of functions with bounded jacobian determinant

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 $$\nabla X_n$$.