Fitting maps of the plane to experimental data

This paper is motivated by the problem of fitting a differential equation to experimental data; a first step is to fit the time 1 map of the corresponding flow, as described in Section 1, and these time 1 maps are the diffeomorphisms in the title of the present paper. The negative part of statement (ii) of Theorem 3.1 implies that there is no algorithm which fits diffeomorphisms of the plane to observed data in a way that depends continuously on the observed data and only on the observed data.