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Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices. (English) Zbl 1066.37052

Lapidus, Michel L. (ed.) et al., Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Analysis, number theory, and dynamical systems. In part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, January 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3637-4/v.1; 0-8218-3292-1/set). Proceedings of Symposia in Pure Mathematics 72, Pt. 1, 155-205 (2004).
Summary: 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 the author [Mém. Soc. Math. Fr., Nouv. Sér. 92 (2003; Zbl 1036.82013] where a renormalization map was introduced in order to describe the spectral properties of self-similar lattices.
Here, 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.
For the entire collection see [Zbl 1055.37002].

MSC:

37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
53D20 Momentum maps; symplectic reduction
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
28A80 Fractals
37E20 Universality and renormalization of dynamical systems
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
78A25 Electromagnetic theory (general)

Citations:

Zbl 1036.82013
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