A note on two types of Lyapunov exponents and entropies for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math>-actions

Zhu, Yong

doi:10.1016/j.jmaa.2017.12.067

Cited by 6 publications

(2 citation statements)

References 27 publications

(47 reference statements)

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“…To investigate the Cellular Automaton map together with the Bernoulli shift, Milnor [17] defined directional entropy. Many of its properties were further studied in [1,3,4,18,19,23].…”

Section: Introductionmentioning

confidence: 99%

Directional Kronecker algebra for $\mathbb{Z}^q$-actions

Liu,

2021

Preprint

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In this paper, directional sequence entropy and directional Kronecker algebra for Z q -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a Z q -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a Z q -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

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“…To investigate the Cellular Automaton map together with the Bernoulli shift, Milnor [17] defined directional entropy. Many of its properties were further studied in [1,3,4,18,19,23].…”

Section: Introductionmentioning

confidence: 99%

Directional Kronecker algebra for $\mathbb{Z}^q$-actions

Liu,

2021

Preprint

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“…entropy formula and SRB measures for random transformations generated by finitely commutative transformations in infinite dimensional Hilbert spaces via its generators, which can be viewed as a generalization of the work in [7,8,24,25] to the infinite dimension spaces. However, the techniques and strategies are completely different due to the feature of infinite dimensional smooth dynamics.…”

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confidence: 99%

Entropies of commuting transformations on Hilbert spaces

Li¹,

Zhu²

2020

Preprint

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By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain C 2 N 2 -actions. . IntroductionThe significance of Pesin's entropy formula (or Ledrappier-Young's entropy formula for SRB measures) lies in its characterizing SRB measures by their Lyapunov exponents and entropy [10]. Pesin's entropy formula for random transformations and stochastic flows of diffeomorphisms in finite dimensional compact spaces were established in [2,11,16,9]. The extension of the above theories to infinite dimensional spaces were

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