Forward Expansiveness and Entropies for Subsystems of Z+k-actions
作者机构：School of Mathematical Sciences Xiamen University
出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报(英文版))
年 卷 期：2023年第39卷第4期
页 面：633-662页
核心收录：
学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学]
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)
主 题：ℤ k + -action forward expansiveness j -dimensional subsystems entropy preimage entropy folding entropy variational principle random transformation 37A35 37B40
摘 要：In this paper,forward expansiveness and entropies of subsystems2) 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.
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