# Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

### Submission summary

 As Contributors: Felix Flicker · Leon Zaporski Arxiv 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 Academic field: Physics Specialties: Dynamical Systems Mathematical Physics 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.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 7, 018 (2019)

### Submission & Refereeing History

#### Published as SciPost Phys. 7, 018 (2019)

Submission 1811.00331v2 on 27 December 2018

## Reports on this Submission

### Anonymous Report 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.

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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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• validity: top
• significance: high
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent