Iterated index and the mean Euler characteristic

作者：Ginzburg Viktor L^*; Goeren Yusuf

来源：Journal of Topology and Analysis, 2015, 7(3): 453-481.

DOI：10.1142/S179352531550017X

• Index and Lefschetz number
• periodic orbits
• contact forms and Reeb flows
• contact and Floer homology
• the mean Euler characteristic

The aim of the paper is three-fold. We begin by proving a formula, both global and local versions, relating the number of periodic orbits of an iterated map and the Lefschetz numbers, or indices in the local case, of its iterations. This formula is then used to express the mean Euler characteristic (MEC) of a contact manifold in terms of local, purely topological, invariants of closed Reeb orbits, without any non-degeneracy assumption on the orbits. Finally, turning to applications of the local MEC formula to dynamics, we use it to reprove a theorem asserting the existence of at least two closed Reeb orbits on the standard contact S-3 (by Cristofaro-Gardiner and Hutchings in the most general form) and the existence of at least two closed geodesics for a Finsler metric on S-2 (Bangert and Long).

• 出版日期2015-9