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Trace maps as 3D reversible dynamical systems with an invariant. (English) Zbl 0830.58025

Summary: One link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so-called trace maps. A subclass of them are mappings on a one-parameter family of \(2D\) surfaces that foliate \(\mathbb{R}^3\) (and also \(\mathbb{C}^3 )\). They are derived from transfer matrix approaches to properties of \(1D\) quasicrystals. In this article, we consider various dynamical properties of trace maps. We first discuss the Fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant \(2D\) surfaces. We highlight a particular surface where the motion is integrable and semiconjugate to an Anosov system (i.e., the mapping acts as a pseudo- Anosov map). We identify properties of symmetry and reversibility (time- reversal symmetry) in the Fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits. We show that a conservative period-doubling sequence can be identified when moving through the one-parameter family of \(2D\) surfaces. By using generator trace maps, in terms of which all trace maps obtained from invertible two-letter substitution rules can be expressed, we show that many features of the Fibonacci trace map hold in general. The role of the Fricke character \(\widehat I(x,y,z) = x^2 + y^2 + z^2 - 2xyz - 1\), its symmetry group, and reversibility for the Nielsen trace maps are described algebraically. Finally, we outline possible higher-dimensional generalizations.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
82D25 Statistical mechanics of crystals
37D99 Dynamical systems with hyperbolic behavior
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