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The ${C}^{1}$ closing lemma, including Hamiltonians. (English) Zbl 0548.58012

Let $\gamma$ be a trajectory of a dynamical system X which is recurrent. Is there a dynamical system close to X with a periodic trajectory close to $\gamma$ ? This question is the closing problem for $\gamma$. As one makes the question precise by assigning a topology to an appropriate space of dynamical systems the solution to this problem can be quite easy or, as is usually the case, difficult in the extreme. Aside from the obvious intrinsic interest of the problem, dynamicists are interested in the closing problem because it is the key ingredient in theorems of general density. For example, the original closing lemma for a recurrent orbit of a ${C}^{1}$ diffeomorphism of a compact manifold due to the first author [Am. J. Math. 89, 956-1009, 1010-1021 (1967; Zbl 0167.218)] implies the general density theorem: If M is a compact manifold then the generic diffeomorphism in the ${C}^{1}$ topology has its periodic points dense in its nonwandering set.

In the paper under review the authors study a new property which they call the lift axiom which is formulated separately for subsets of diffeomorphisms, flows and vector fields. For diffeomorphisms one considers a subset S of the ${C}^{1}$ diffeomorphisms of a compact Riemannian manifold with exponential map exp which imbeds each unit ball in the unit sphere bundle into M. S satisfies the lift axiom if for each $f\in S$ and each ${C}^{1}$ neighborhood U of f there is an $ϵ>0$ such that whenever V is a unit tangent vector at p there is a diffeomorphism close to the identity satisfying $g\circ f\in U$ and (L1) $g\left(p\right)=exp\left(ϵv\right)$, (L2) the set of all points where g is not the identity is contained in ${exp}_{p}\left({T}_{p}M\left(r\right)\right)$, the exponential image of the unit r ball and (L3) if ${g}_{1},···,{g}_{n}$ are several such perturbations with disjoint support then ${g}_{1}\circ ···\circ {g}_{n}\circ f\in S$. The main result is the following: if S satisfies the lift axiom then S has the closing property. This result is used to prove the original closing lemma and the ${C}^{1}$ case of Poincaré’s conjecture on Hamiltonian vector fields that ${C}^{r}$ generically in the space of Hamiltonian vector fields the periodic trajectories are dense in the compact energy surfaces.

The paper contains an informative introduction which is recommended even to the nonspecialist as a precise account of the results of the paper; see also the earlier paper of the second author [Lect. Notes Math. 668, 225-230 (1978; Zbl 0403.58020)].

Reviewer: C.Chicone

##### MSC:
 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 37C10 Vector fields, flows, ordinary differential equations 58A10 Differential forms (global analysis)