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Generic unfoldings with the same bifurcation diagram which are not \((C^0,C^0)\)-equivalent. (English) Zbl 0901.58033

This paper deals with germs at 0 of \(k\)-parameter families of \(C^\infty\) diffeomorphisms \(f_\mu (x)\) on the line. Two families \(f_\mu\) and \(g_\nu\) are \((C^r, C^0)\)-conjugate if there exist a \(C^r\) diffeomorphism \(\varphi\) on \(\mathbb{R}^k,0\) and for any \(\mu\) a homeomorphism \(h_\mu\) on \(\mathbb{R},0\) conjugating \(f_\mu\) and \(g_{\nu= \varphi (\mu)}\). The case \(r=0\) is considered and the notion of \((C^r, C^0)\)-equivalence is (in a similar way) also discussed.
First of all, the authors derive some properties of the Mather invariant for real diffeomorphisms. Then they use this invariant to construct infinitely many families \(F_\zeta\) \((\zeta \in \Lambda)\) such that \(F_{\zeta_1}\), \(F_{\zeta_2}\) are not \((C^0, C^0)\)-conjugate provided that \(\zeta_1 \neq \zeta_2\).
The following theorems are stated in the paper:
Theorem 1.1. There exists a set parametrized by a functional space of 2 by 2 non-\((C^0, C^0)\)-conjugate generic 3-parameter local families.
Theorem 1.2. There exists a set parametrized by a functional space of 2 by 2 non-\((C^0, C^0)\)-equivalent generic 4-parameter unfoldings of Hopf-Takens type.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] Mather, J., Commutators of diffeomorphisms, Comm. Math. Helv., vol. 48, 195-233 (1973)
[2] Roussarie, R.; Camacho, M. I.; Pacifico, M. J.; Takens, F., Weak and continuous equivalences for families of line diffeomorphisms, Dynamical systems and bifurcation theory. Dynamical systems and bifurcation theory, Pitman Research Notes in Mathematics series, 160, 377-385 (1987)
[3] Takens, F., Unfoldings of certain singularities of vector fields, Generalized Hopf bifurcations. Generalized Hopf bifurcations, J.D.E., \(n^o 14, 476-493 (1973)\) · Zbl 0273.35009
[4] Yoccoz, J.-Ch., Petits diviseurs en dimension 1, Astérisque, vol. 231 (1996)
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