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Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
by Leon Zaporski, Felix Flicker
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Submission summary
| Authors (as registered SciPost users): | Felix Flicker · Leon Zaporski |
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| Preprint Link: | https://arxiv.org/abs/1811.00331v2 (pdf) |
| Date accepted: | 2019-08-05 |
| Date submitted: | 2018-12-27 01:00 |
| Submitted by: | Zaporski, Leon |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.
Published as SciPost Phys. 7, 018 (2019)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2019-7-31 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:1811.00331v2, delivered 2019-07-31, doi: 10.21468/SciPost.Report.1087
Strengths
Substantial new results on symbolic dynamics of substitution sequences, especially in relation to quasilattices.
Weaknesses
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Report
This paper contributes to the field of symbolic dynamics of substitution sequences. It provides ample motivation, for example drawing from the emergence of quasilattices, in this context termed time quasilattices. The main theme of the paper is the quantitative study of the complexity of a variety of symbolic sequences, through the study of their topological entropy. This quantity is zero for relatively simple period-doubling cascades but increase monotonically for more general sequences. The paper presents a thorough study, providing analytical and numerical evidence for the convergence of this entropy on trajectories linear on the ordinal $n$ of a word within a sequence. The paper also establishes that all 10 classes of the Boyle-Steinhardt classification of quasilattices can be realized through substitution dynamics.
Requested changes
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