Directional Preimage Entropy for $\mathbb{Z}_+^k$-Actions
Received:November 12, 2018  Revised:September 04, 2019
Key Word: $\mathbb{Z}_{+}^k$-actions   directional preimage entropy   infinite graph  
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Author NameAffiliation
Ya-na GAO Faculty of International Language Education, Hebei Foreign Studies University, Hebei 050000, P. R. China 
Ziyao ZHANG College of Mathematics and Computer Science, Hanjiang Normal University, Hubei 442000, P. R. China 
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      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.
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