Amorphic complexity
| Autor: | Gabriel Fuhrmann, Tobias Jäger, Maik Gröger |
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| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
Dynamical systems theory Applied Mathematics 010102 general mathematics General Physics and Astronomy Statistical and Nonlinear Physics Context (language use) Dynamical Systems (math.DS) Equicontinuity 01 natural sciences Toeplitz matrix Group action 0103 physical sciences Attractor FOS: Mathematics Entropy (information theory) 010307 mathematical physics 0101 mathematics Invariant (mathematics) Mathematics - Dynamical Systems Mathematical Physics Mathematics |
| Zdroj: | Nonlinearity |
| DOI: | 10.48550/arxiv.1503.01036 |
| Popis: | We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems. After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work. Comment: 36 pages, 2 figures, added references, minor changes, corrected typos |
| Databáze: | OpenAIRE |
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