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Some remarks on modified power entropy

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M. Gröger, T. Jäger

Dynamical Systems · Submitted 23 Jun 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems.

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arXiv ID1506.07192 (version 1)
Pith IDWWZHPGTX, Pith's permanent record of this paper and its review
CitationM. Gr\"oger, T. J\"ager. "Some remarks on modified power entropy." arXiv:1506.07192 (2015). https://pith.science/paper/WWZHPGTX
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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.92 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

The aim of this note is to point out some observations concerning modified power entropy of $\Z$- and $\N$-actions. First, we provide an elementary example showing that this quantity is sensitive to transient dynamics, and therefore does not satisfy a variational principle. Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems. In this respect, it differs from power entropy and amorphic complexity, which are two further topological invariants for zero entropy systems (`slow entropies'). Finally, we construct an example of an irregular Toeplitz flow with zero modified power entropy.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The aim of this note is to point out some observations concerning modified power entropy of $\Z$- and $\N$-actions. contribution 0.84 0.36 0.07 asserted 0.55
s1 First, we provide an elementary example showing that this quantity is sensitive to transient dynamics, and therefore does not satisfy a variational principle. result 0.82 0.95 0.23 asserted 1.00
s2 Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems. result 0.96 0.91 0.26 asserted 1.00
s3 In this respect, it differs from power entropy and amorphic complexity, which are two further topological invariants for zero entropy systems (`slow entropies'). interpretation 0.73 0.25 0.15 asserted 1.00
s4 Finally, we construct an example of an irregular Toeplitz flow with zero modified power entropy. result 0.64 0.96 0.14 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.61 · evidence from proof or derivation 0.96.

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The paper
Original abstract

The aim of this note is to point out some observations concerning modified power entropy of $\Z$- and $\N$-actions. First, we provide an elementary example showing that this quantity is sensitive to transient dynamics, and therefore does not satisfy a variational principle. Further, we show that modified power entropy is not suitable to detect the break of equicontinuity which takes place during the transition from almost periodic to almost automorphic minimal systems. In this respect, it differs from power entropy and amorphic complexity, which are two further topological invariants for zero entropy systems (`slow entropies'). Finally, we construct an example of an irregular Toeplitz flow with zero modified power entropy.

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1 arXiv paper cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 8 sentences without a confident label

Builds on 1 paper
  • The main reason for treating amorphic complexity here is to complete the discussion in [ref] by showing that there are no direct relations, in terms of an inequality, between amorphic complexity and the other two notions.

    Amorphic complexity · Power entropy, modified power entropy and amorphic complexity · 0.80

Compares with 1 paper
  • Theorem [ref] is in contrast to the following result in [ref] , which shows that the asymptotic separation numbers involved in the definition of amorphic complexity are sensitive to the break of equicontinuity in the above situation.

    Amorphic complexity · Modified power entropy of regular almost 1-1 extensions · 0.98

Cites as background 2 papers
  • Basic properties of this quantity, like topological invariance, factor relations, power invariance and a product rule, as well as the application to a number of example classes are discussed in [ref] .

    Amorphic complexity · Power entropy, modified power entropy and amorphic complexity · 0.95

  • However, we will not go into any detail in this direction and refer to [ref] for a discussion and further references.

    1406.6423 · Some thoughts on slow entropies · 1.00

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