A Bangert-Hingston Theorem for Starshaped Hypersurfaces

Abstract

Let Q be a closed manifold with non-trivial first Betti number and Σ ⊆ T * Q a generic starshaped hypersurface. We prove that the number of geometrically distinct Reeb orbits of period at most T on Σ grows at least logarithmically in T whenever Q admits a nontrivial S 1 -action.then the spectral invariants are related to each other viaThis is based on a pinching argument due to Macarini and Schlenk [MS11], which also plays a key role in the arguments of the aforementioned papers [Hei11; MMP12; Wul14]. The last ingredient is the use of a Robbin-Salamon index growth result due to M. de Gosson, S. De Gosson and Piccione [DGDGP08] together with an iteration argument that is very much inspired by Irie's work [Iri14].Under the S 1 -action assumption, Irie's Main Theorem [Iri14] is applicable, however this is not particularly fruitful when applied to F due to homogeneity -while the finiteness of the Hofer-Zehnder capacity of the disk cotangent bundle D * Q implies dense existence of orbits nearby Σ = F −1 (1) [HZ12], it could happen that these orbits all correspond to a single Reeb orbit on Σ. To make matters worse, no growth rate control can be extrapolated even if we knew that there are infinitely many Reeb orbits. In this sense, the Main Theorem can be viewed as a strengthening to the dynamical consequences of Irie's Theorem in the case of starshaped hypersurfaces.Let us mention that there is no need to bound the pinching factor σ in our arguments. This seems to be a reoccurring dichotomy between starshaped hypersurfaces in T * Q (with Q closed) and (compact) starshaped hypersurfaces in R 2n , where for the latter the pinching factor σ plays a more crucial role [Gir84; Ber+85; Vit89; Eke12; AGH16; Wan16; DL17; AM17].The methods in BH (Bangert and Hingston) [BH84] are not well suited to our setting since spectral invariants can only be assigned to homology classes, while in the Lagrangian case, as done in BH, minimax values can be assigned to homotopy classes as well. This issue can be ascribed to the lack of a global flow for the negative gradient of the Hamiltonian action functional, which is in contrast to the existence of a global flow of the negative gradient of the Lagrangian action functional [Oh02]. Even under further assumptions, e.g. that the relevant homotopy groups inject into the free loop space homology via the Hurewicz homomorphism, we have not been able to adapt BH's arguments.Remark 1.1. In the Σ convex case the Lagrangian formulation is sufficient to run the proof strategy of [BH84], also see [BJ08] for the Finsler case. This gives a better growth rate, i.e. T log(T ), without any genericity assumptions. The present work will be part of the author's PhD thesis, in which we will address the case of Σ convex without the generic assumption -this gives some further insights on the key differences between starshapedness and convexity.

A Bangert–Hingston theorem for starshaped hypersurfaces

“…This follows verbatim from [36, page 431] and/or [27, page 2491]-the non-compactness does not affect the proof due to the imposed C 0 -bound on the difference H 0 − H 1 . We refer the reader to [34] for a full proof.…”