电网络、辛约化及其在自相似格重整化映射中的应用
Electrical Networks, Symplectic Reductions, and Application to the Renormalization Map of Self-Similar Lattices
摘要：
本文的第一部分涉及电网络和辛约化。我们考虑两个操作电网络（“迹映射”和“胶合映射”），并表明它们对应于辛约化。我们也给出了辛约化的一些一般性质，特别是我们研究了辛约化作为Lagrange Grassmannian上的有理映射时的奇异性。这是由[23]激发的，其中引入了重整化映射以描述自相似晶格的谱特性。在本文中，我们表明，这个重整化映射可以表示在辛减少，它的一些关键属性是辛减少的一般性质的直接后果（和辛减少的奇异性发挥重要作用，与我们的运营商的谱性质）。我们还提出了新的例子，我们可以计算重整化映射。
英文摘要：
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.
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