On the concept of integrability for discrete dynamical systems. Investigation of wandering points of some trace map. (English) Zbl 1321.37020
López-Ruiz, Ricardo (ed.) et al., Nonlinear maps and their applications. Selected contributions from the NOMA 2013 international workshop, Zaragoza, Spain, September 3–4, 2013. Cham: Springer (ISBN 978-3-319-12327-1/hbk; 978-3-319-12328-8/ebook). Springer Proceedings in Mathematics & Statistics 112, 127-158 (2015).
Summary: We extend the concept of integrability suggested by R.I. Grigorchuk for a polynomial discrete dynamical system to an arbitrary discrete dynamical system in the plane. This extension makes it possible to reduce an integrable dynamical system to a dynamical system of the skew products class.
We formulate and prove the criterion for integrability. As the first step of the investigation of the nonwandering set of the trace map \(F(x, y)=(xy, (x-2)^2)\), which arises in quasicrystal physics, we describe some geometric constructions. We prove that all points of constructed set are wandering.
For the entire collection see [Zbl 1309.37004].
We formulate and prove the criterion for integrability. As the first step of the investigation of the nonwandering set of the trace map \(F(x, y)=(xy, (x-2)^2)\), which arises in quasicrystal physics, we describe some geometric constructions. We prove that all points of constructed set are wandering.
For the entire collection see [Zbl 1309.37004].
MSC:
| 37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
| 34C14 | Symmetries, invariants of ordinary differential equations |
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