Multiple reductions, foliations and the dynamics of cluster maps

TL;DR

This paper explores the geometric reduction of cluster maps using presymplectic and Poisson foliations, providing a unified framework to understand their dynamics and multiple reduction scenarios with detailed examples.

Contribution

It introduces a geometric foliation-based approach to reduce and analyze cluster map dynamics, including cases with multiple reductions and complex examples.

Findings

Unified geometric description of cluster map reductions

Analysis of multiple reduction scenarios

Detailed example in seven dimensions

Abstract

Presymplectic and Poisson reduction of cluster maps are described in terms of the "canonical" foliations of presymplectic and Poisson manifolds. This approach to reduction leads to a geometric description, in terms of foliations, of the dynamics of the original (not reduced) map. The case where multiple reductions exist (presymplectic/Poisson or presymplectic/Poisson/Poisson) is further explored and examples illustrating several features of this approach are presented, including a nontrivial one in dimension seven which is comprehensively treated.