Attractor Networks on Complex Flag Manifolds

TL;DR

This paper explores attractor-repellor networks on complex flag manifolds, revealing their lattice structure and connection to the Bruhat order, with new proofs and elementary dynamical systems approach.

Contribution

It provides explicit examples of attractor-repellor networks on complex flag manifolds, linking the Smale order to the Bruhat order with a novel proof using the Lambda-Lemma.

Findings

The Smale order on fixed points corresponds to the Bruhat order.

Explicit determination of attractor-repellor networks on complex flag manifolds.

New proof of the order relation using the Lambda-Lemma of Palis.

Abstract

Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that the Smale order on the set of fixed points can be identified with the well-known Bruhat order. This could also be derived from results of Kazhdan and Lusztig, but we give a new proof using the Lambda-Lemma of Palis. For the convenience of the reader we also introduce the flag manifolds via elementary dynamical systems using only a minimum of Lie theory.