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Forward Expansiveness and Entropies for Subsystems of Z+k-actions

Acta Mathematica Sinica,English Series 2023年第4期39卷 633-662页

作者： Yao Jia GUO Xin Sheng WANG Yu Jun ZHU School of Mathematical Sciences Xiamen University

In this paper,forward expansiveness and entropies of "subsystems"2) of Z+k-actions are investigated.Let α be a Z+k-action on a compact metric space.For each 1 ≤j≤k-1,denote Gj+={V+:=V ∩R+k:V is a j-dimensional subspace of Rk}.We consider the forward expansiveness and entropies for α along V+∈ Gj+.Adapting the technique of "coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Zk-actions,to the Z+kcases,we show that the set Ej+(α) of forward expansive j-dimensional V+is open in Gj+.The topological entropy and measure-theoretic entropy of j-dimensional subsystems of α are both continuous in Ej+(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L ∈G1+,we relate the 1-dimensional subsystem of α along L to an i.***.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions) via the preimage structure of α along L.We show that the preimage entropy coincides with the classical entropy along any L ∈E1+(α) for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.

关键词： ℤ k + -action forward expansiveness j -dimensional subsystems entropy preimage entropy folding entropy variational principle random transformation 37A35 37B40

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On an entropy of Zk＋-Actions

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Acta Mathematica Sinica,English Series 2014年第3期30卷 467-480页

作者： Yu Jun ZHU Wen Da ZHANG College of Mathematics and Information Science and Hebei Key Laboratory of Computational Mathematics and ApplicationsHebei Normal University College of Mathematics and Information Science Hebei Normal University

In this paper,a definition of entropy for Z＋k（k≥2）-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite（even zero） entropy as single transformations.Some basic properties are investigated and its value for the Z＋k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z＋k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the ＂inverse orbits＂ structure of the system.

关键词： entropy preimage entropy Zk＋-action

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Forward Expansiveness and Entropies for Subsystems of Z^(k)_(+)-actions

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Acta Mathematica Sinica,English Series 2023年第4期39卷 633-662页

作者： Yao Jia GUO Xin Sheng WANG Yu Jun ZHU School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.***.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.

关键词： Z^(k)_(+)-action forward expansiveness j-dimensional subsystems entropy preimage entropy folding entropy variational principle random transformation

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