## The $$C^1$$ closing lemma. (Le “closing lemma” en topologie $$C^1$$.)(French)Zbl 0920.58039

The $$C^1$$ closing lemma asserts that, given a dynamical system $$f$$ in a compact manifold $$M$$ and a recurrent point $$p$$ of $$f$$, there exists a dynamical system $$g$$ in any neighborhood of $$f$$ in the $$C^1$$ topology such that $$p$$ is a periodic point of $$g$$. The lemma was first proved by C. C. Pugh [Am. J. Math. 89, 956-1009 (1967; 167.21803)]. The proof was very complicated and had a minor error. An improved proof was given by C. C. Pugh and C. Robinson [Ergodic Theory Dyn. Syst. 3, 261-313 (1983; 548.58012)]. But their proof is still hard. Mai gave a simpler proof of the lemma based on Liao’s techniques and his own ones of movements under a sequence of linear transformations [J.-H. Mai, Sci. Sin., Ser. A 29, 1020-1031 (1986; Zbl 0616.58024); Chin. Sci. Bull. 34, 179-184 (1989; Zbl 0686.58030)]. Mañé studied the stability properties of discrete dynamical systems, i.e., $$C^1$$-diffeomorphisms of closed manifolds [Ann. Math. (2) 116, 503-540 (1982; Zbl 0511.58029)].
In the paper under review, the author uses Mai’s algebraic result to simplify the proof of the $$C^1$$ closing lemma and solves a new case of symplectic fields. Moreover, the author generalizes the $$C^1$$ ergodic closing lemma of Mañé (the ergodic version of the orbit closing lemma) to the Borelian positive measures, finite on every compact, defined on noncompact manifolds.

### MSC:

 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37G99 Local and nonlocal bifurcation theory for dynamical systems 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37A99 Ergodic theory 37C10 Dynamics induced by flows and semiflows
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### References:

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