Non-stabilisable jets of diffeomorphisms in \({\mathbb{R}}^ 2\) and of vector fields in \({\mathbb{R}}^ 3\) (Q1093225)

The following two results are proved. Theorem 1. For any X, germ of a \(C^{\infty}\) vector field on \({\mathbb{R}}^ 2\), with a certain specific 9- jet, there is a germ of a \(C^{\infty}\) diffeomorphism f with \(j_{\infty}(f)(0)=j_{\infty}(X_ 1)(0)\), \(X_ t\) denoting the flow of X, such that f and \(X_ 1\) are not \(C^ 0\)-conjugate. Theorem 2. Whenever X is a germ of a \(C^{\infty}\) vector field on \({\mathbb{R}}^ 3\) whose 9-jet belongs to a certain class of 9-jets, there can be found a germ of a \(C^{\infty}\) vector field Y on \({\mathbb{R}}^ 3\) with \(j_{\infty}(X)(0)=j_{\infty}(Y)(0)\) such that X and Y are not \(C^ 0\)-equivalent. The second result implies the existence of jets of vector fields in \({\mathbb{R}}^ 3\) which are not stabilizable for the notion of \(C^ 0\)- equivalence. The question of the existence of such jets was raised by R. Thom in 1970 and very soon \textit{F. Takens} found examples in \({\mathbb{R}}^ 4\) with \(n\geq 4\) [Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 583-597 (1973; Zbl 0283.58009); Publ. Math., Inst. Hautes Étud. Sci. 43(1973), 47-100 (1974; Zbl 0279.58009)]. A short survey of the construction exposed in this paper can be found in the author's earlier paper [Lect. Notes Math. 1125, 15-46 (1985; Zbl 0575.58003)].