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Eurasian Journal of Mathematical and Computer Applications, 2018 Volume 6, Issue 3, Pages 4–33 (Mi ejmca118)

This article is cited in 1 paper

Eikonal algebra on a graph of simple structure

M. I. Belishevab, A. V. Kaplunb

a Saint-Petersburg Department of the Steklov Mathematical Institute, RAS, 27 Fontanka, St Petersburg 191023, Russia
b Saint-Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

Abstract: An eikonal algebra $\mathfrak{E}(\Omega)$ is a $C^*$-algebra related to a metric graph $\Omega$. It is determined by trajectories and reachable sets of a dynamical system associated with the graph. The system describes the waves, which are initiated by boundary sources (controls) and propagate into the graph with finite velocity. Motivation and interest to eikonal algebras comes from the inverse problem of reconstruction of the graph via its dynamical and/or spectral boundary data. Algebra $\mathfrak{E}(\Omega)$ is determined by these data. In the mean time, its structure and algebraic invariants (irreducible representations) are connected with topology of $\Omega$. We demonstrate such connections and study $\mathfrak{E}(\Omega)$ by the example of $\Omega$ of a simple structure. Hopefully, in future, these connections will provide an approach to reconstruction.

Keywords: metric graph, hyperbolic dynamical system, reachable sets, $C^*$-algebra of eikonals.

MSC: 46Lxx, 34B45, 35Qxx, 35R30

Received: 01.03.2018
Accepted: 02.07.2018

Language: English

DOI: 10.32523/2306-6172-2018-6-3-4-33



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