Diffeomorphisms of the closed unit disc converging to the identity

Abstract

If $\mathcal{G}$ is the group (under composition) of diffeomorphisms $f:\bar{D}(0;1)\to \bar{D}(0;1)$ of the closed unit disc $\bar{D}(0;1)$ which are the identity map $id:\bar{D}(0;1)\to \bar{D}(0;1)$ on the closed unit circle and satisfy the condition $det(J(f))>0$, where $J(f)$ is the Jacobian matrix of $f$ or (equivalently) the Fréchet derivative of $f$, then $\mathcal{G}$ equipped with the metric $d_{\mathcal{G}}(f,g)=\Vert f-g{\Vert }_{\infty }+\Vert J(f)-J(g){\Vert }_{\infty }$, where $f$, $g$ range over $\mathcal{G}$, is a metric space in which $d_{\mathcal{G}}\left(f_t,id\right)\to 0$ as $t\to 1^{+}$, where $f_t(z)=\frac{tz}{1+(t-1)|z|}$, whenever $z\in \bar{D}(0;1)$ and $t\ge 1$.