Decomposing diffeomorphisms of the sphere (Q2893277)

A diffeomorphism of the sphere \(S^n\) cannot, in general, be decomposed into diffeomorphisms that are \(C^1\) close to the identity -- this is due the existence of exotic spheres in high dimensions, see [\textit{J. W. Milnor}, Ann. Math. (2) 64, 399--405 (1956; Zbl 0072.18402)]. The authors show that for every \(\varepsilon > 0\) a diffeomorphism \(f : S^n \rightarrow S^n\) can be decomposed as \(f = f_m \circ \dots \circ f_1\), where \(f_k\) is \((1+\varepsilon)\)-bilipschitz with respect to the spherical metric \(q\) on \(S^n\) and \(q(f_k(x), x) < \varepsilon\) for all \(x \in S^n\). A byproduct of the construction is that every \(f_k\) is differentiable but need not be continuously differentiable. The proof also shows that \(f\) can be connected to the identity by a path put together of special bilipschitz paths. Contrary to the diffeomorphism case little is known about factorization of bilipschitz mappings; see [\textit{M. Freedman} and \textit{Z.-X. He}, Invent. Math. 92, No. 1, 129--138 (1988; Zbl 0622.30011)] and [\textit{V. Gutlyanskii} and the reviewer, Conform. Geom. Dyn. 5, 6--20 (2001; Zbl 0981.30014)]. For the proof the authors make use of the Munkres deformation result for \(C^k\)-diffeomorphisms, see [\textit{F. W. Wilson jun.}, Ill. J. Math. 16, 222--233 (1972; Zbl 0238.57016)].