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

Abstract

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 $\mathbb{G}_j^ + = \left\{ {{V_ + }: = V \cap \mathbb{R}_ + ^k:V} \right.$ is a j-dimensional subspace of ℝ^k. We consider the forward expansiveness and entropies for α along ${V_ + } \in \mathbb{G}_j^ +$. 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 $\mathbb{E}_j^ + \left( \alpha \right)$ of forward expansive j-dimensional V₊ is open in $\mathbb{G}_j^ +$. The topological entropy and measure-theoretic entropy of j-dimensional subsystems of α are both continuous in $\mathbb{E}_j^ + \left( \alpha \right)$, and moreover, a variational principle relating them is obtained.

For a 1-dimensional ray $L \in \mathbb{G}_1^ +$, we relate the 1-dimensional subsystem of α along 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 of α along L. We show that the preimage entropy coincides with the classical entropy along any $L \in \mathbb{E}_1^ + \left( \alpha \right)$ 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.