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**The dynamics on three-dimensional strictly convex energy surfaces.**
*(English)*
Zbl 0944.37031

This is an impressive piece of mathematics which to a large extent is based on the theory of pseudoholomorphic curves, developed by the same authors in previous work.

Consider \({\mathbb R}^4\) with the standard symplectic form \(\omega = d\lambda_0\) where \(\lambda_0 = {1\over 2} (q_j dp_j - p_j dq_j)\), and let \(S\) be a compact energy surface \(H(x)=1\) of a regular Hamiltonian. Assuming \(H''(x) \geq a \text{ Id}\), the energy surface (which is assumed to contain the origin in its interior) is strictly convex. Such hypersurfaces \(S\) are known to carry at least 2 periodic orbits. One of the main theorems in this work states that there are either 2 or infinitely many such orbits. The real “tour de force” (from which this result is deduced) is the explicit construction of a global surface of section of \(S\), i.e. an embedded compact surface whose boundary components consist of periodic orbits and whose interior is suitably transversal to the flow. The desired surface of section requires singling out a special periodic solution, based on an index concept.

Section 3 contains a detailed account of this theory and proves the following fundamental result. On a 3-dimensional manifold with contact form \(\lambda\) and associated Reeb vector field \(X_\lambda\), every contractible \(T\)-periodic solution \(x\) of \(X_\lambda\) gives rise to an integer \(\widetilde{\mu}(x,T)\); in the special case of a strictly convex hypersurface \(S\) of \({\mathbb R}^4\), equipped with the contact form \(\lambda_0|_S\), one has \(\widetilde{\mu}(x,T) \geq 3\). Since the property of being strictly convex is not symplectically invariant, the authors introduce the notion of a dynamically convex contact form \(\lambda\) on \(S^3\), which is one with the property \(\widetilde{\mu}\geq 3\) for every \(T\)-periodic solution of \(X_\lambda\). Assuming \(\lambda = f \lambda_0\) is dynamically convex (where \(f\) is a smooth positive function on \(S^3\)), the special periodic solution which is going to be used as a spanning orbit for a global surface of section of disc type, is one which has the property \(\widetilde{\mu}=3\). The alternative “2 or \(\infty\)” for the number of periodic solutions is then a consequence of a theorem of Franks which states that if an area-preserving homeomorphism of an open annulus admits one periodic orbit, it has infinitely many such orbits.

Section 2 gives an outline of the construction of a surface of section and states as main theorem the existence of an “open book decomposition” of the 3-manifold (at least in a special case).

Sections 4 and 5 contain the hard work: the proofs of all main theorems, under the additional assumption that the periodic orbits under consideration have a nondegeneracy property. In the last three sections, this nondegeneracy requirement is eliminated by an approximation construction.

Consider \({\mathbb R}^4\) with the standard symplectic form \(\omega = d\lambda_0\) where \(\lambda_0 = {1\over 2} (q_j dp_j - p_j dq_j)\), and let \(S\) be a compact energy surface \(H(x)=1\) of a regular Hamiltonian. Assuming \(H''(x) \geq a \text{ Id}\), the energy surface (which is assumed to contain the origin in its interior) is strictly convex. Such hypersurfaces \(S\) are known to carry at least 2 periodic orbits. One of the main theorems in this work states that there are either 2 or infinitely many such orbits. The real “tour de force” (from which this result is deduced) is the explicit construction of a global surface of section of \(S\), i.e. an embedded compact surface whose boundary components consist of periodic orbits and whose interior is suitably transversal to the flow. The desired surface of section requires singling out a special periodic solution, based on an index concept.

Section 3 contains a detailed account of this theory and proves the following fundamental result. On a 3-dimensional manifold with contact form \(\lambda\) and associated Reeb vector field \(X_\lambda\), every contractible \(T\)-periodic solution \(x\) of \(X_\lambda\) gives rise to an integer \(\widetilde{\mu}(x,T)\); in the special case of a strictly convex hypersurface \(S\) of \({\mathbb R}^4\), equipped with the contact form \(\lambda_0|_S\), one has \(\widetilde{\mu}(x,T) \geq 3\). Since the property of being strictly convex is not symplectically invariant, the authors introduce the notion of a dynamically convex contact form \(\lambda\) on \(S^3\), which is one with the property \(\widetilde{\mu}\geq 3\) for every \(T\)-periodic solution of \(X_\lambda\). Assuming \(\lambda = f \lambda_0\) is dynamically convex (where \(f\) is a smooth positive function on \(S^3\)), the special periodic solution which is going to be used as a spanning orbit for a global surface of section of disc type, is one which has the property \(\widetilde{\mu}=3\). The alternative “2 or \(\infty\)” for the number of periodic solutions is then a consequence of a theorem of Franks which states that if an area-preserving homeomorphism of an open annulus admits one periodic orbit, it has infinitely many such orbits.

Section 2 gives an outline of the construction of a surface of section and states as main theorem the existence of an “open book decomposition” of the 3-manifold (at least in a special case).

Sections 4 and 5 contain the hard work: the proofs of all main theorems, under the additional assumption that the periodic orbits under consideration have a nondegeneracy property. In the last three sections, this nondegeneracy requirement is eliminated by an approximation construction.

Reviewer: Willy Sarlet (Gent)