Directional Preimage Entropy for $\mathbb{Z}_+^k$-Actions

Abstract:

      In this paper, a new type of entropy, directional preimage entropy including topological and measure theoretic versions for $\mathbb{Z}_{+}^k$-actions, is introduced. Some of their properties including relationships and the invariance are obtained. Moreover, several systems including $\mathbb{Z}_{+}^k$-actions generated by the expanding maps, $\mathbb{Z}_{+}^k$-actions defined on finite graphs and some infinite graphs with zero directional preimage branch entropy are studied.