Electrical Networks, Symplectic Reductions, and Application to the Renormalization Map of Self-Similar Lattices

TL;DR

This paper explores the connection between electrical networks, symplectic reductions, and their application to the renormalization map of self-similar lattices, revealing new insights into spectral properties and singularities.

Contribution

It demonstrates that the renormalization map for self-similar lattices can be formulated using symplectic reductions, linking spectral analysis with symplectic geometry.

Findings

Renormalization map expressed via symplectic reductions

Singularities influence spectral properties

New examples of computable renormalization maps

Abstract

The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the "trace map" and the "gluing map") and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of self-similar lattices. In this text, we show that this renormalization map can be expressed in terms of symplectic reductions and that some of its key properties are direct consequences of general properties of symplectic reductions (and the singularities of the symplectic reduction play an important role in relation with the spectral properties of our operator). We also present new examples where we can compute the renormalization map.