Diffeomorphisms of the closed unit disc converging to the identity

TL;DR

This paper studies the convergence of a family of diffeomorphisms of the closed unit disc to the identity map, focusing on the metric space structure and the behavior of specific transformations.

Contribution

It introduces a metric on the group of certain diffeomorphisms and proves their convergence to the identity as a parameter approaches 1 from above.

Findings

The group of diffeomorphisms forms a metric space under the defined metric.

The family of maps $f_t$ converges to the identity as $t o 1^+$.

The convergence is characterized in terms of the supremum norm and Jacobian differences.

Abstract

If $\mathcal{G}$ is the group (under composition) of diffeomorphisms $f : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ of the closed unit disc ${\bar{D}}(0;1)$ which are the identity map $id : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ 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 Fr\'echet 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 $d_{\mathcal{G}} \left( f_{t} , id \right) \rightarrow 0$ as $t \rightarrow 1^{+}$, where $f_{t}(z) = \frac{ tz }{ 1 + (t-1) \vert z \vert }$, whenever $z \in {\bar{D}}(0;1)$ and $t \geq 1$.