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

作　　者：Yao Jia GUO Xin Sheng WANG Yu Jun ZHU 

机构地区：[1]School of Mathematical Sciences,Xiamen University,Xiamen 361005,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2023年第4期633-662,共30页数学学报（英文版）

基　　金：Wang and Zhu are supported by NSFC (Grant Nos.11771118,11801336,12171400);Wang is also supported by China Postdoctoral Science Foundation (No.2021M691889)。

摘　　要：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.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 

分 类 号：O189.11[理学—数学]