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).

Abstract-A number of switch fabric architectures based on mini-router grids (MRG) have been proposed as a replacement of buses for system-on-chip communication, as well as a replacement of crossbars for network routers. The rationale for using MRGs in switch fabrics is that they provide high delivery ratios, low latencies, high degree of parallelism and pipelining, load balancing properties, and sub-quadratic cost growth for their implementation. The traditional approaches to switch fabrics are based on input queuing (IQ) or virtual output queueing (VOQ), because output queuing (OQ) solutions to date are unscalable and expensive due to the speedup problem. However, we show that the speedup problem introduced by OQ can be bounded by 3 by using MRGs.We present the design of a switch fabric based on OQ MRGs that offers high delivery ratios, smaller queue sizes, and QoS guarantees. Queueing and scheduling are distributed over the MRs, where each MR is a pipestage, thus allowing MRGs to provide high throughput by nature. We present the first in-depth analytical model of switch fabric architectures based on OQ MRG, and support our model with register-transfer level (RTL) simulations in SystemC. The analytical and simulation results are shown to have close correlation over a range of design parameters and evaluation metrics.

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