Meaning of $h_*$ complex map.

Question

I have come across the following:

Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and let $h: \Omega \rightarrow \mathbb{C}^n$ (or $\mathbb{R}^n$) be a smooth mapping with $||h_*||= sup_x || h_*(x)|| < 1$, then the mapping $H$ taking $x \mapsto x + h(x)$ is a diffeomorphism of $\Omega$ onto $H\Omega$.

What does $h_*$ stand for in this context?

Answer 1

From context it has to be the induced map of the tangent bundles, in other words, the Jacobian matrix $h_*=Dh=h'=J_h$. Then the claim is just a consequence of the inverse function theorem, as the derivative of $x+h(x)$, which is $I+h'(x)$, is invertible everywhere with $\|(I+h'(x))^{-1}\|\le(1-\|h_*\|)^{-1}$.