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Non-stabilisable jets of diffeomorphisms in $${\mathbb{R}}^ 2$$ and of vector fields in $${\mathbb{R}}^ 3$$. (English) Zbl 0628.58037
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 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)].