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I read the paper Entropie d'un groupe abélien de transformation by Conze and the part of the book Dynamical systems of Algebraic Origin by Schmidt about the entropy for $\mathbb{Z}^d$ actions. I was wondering if it is possible to generalize the definition of topological and measure-theoretic entropy to $\mathbb{N}^d$ actions using the same idea. If so, what happens if we compute the entropy of a $\mathbb{Z}^d$ action using the definition for $\mathbb{N}^d$? Do we get the same result?

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The extension of both topological and measure entropy to $\mathbb N^d$-actions is straightforward (in the case of measure entropy you just need to remember that you are taking inverse images of sets, and so the maps do not need to be invertible). Another, equivalent, approach is to take the "natural extension" of the action, using inverse limits, to obtain a $\mathbb Z^d$-action with the same entropy. If you start with a $\mathbb Z^d$-action and think of it as an $\mathbb N^d$-action, the natural extension is just the $\mathbb Z^d$-action, so has the same entropy.

More interesting is the problem of isomorphism, which can lead to quite different answers in the two cases. If a $\mathbb Z^d$-action is considered as an $\mathbb N^d$-action, then an isomorphism can only depend on coordinates in $\mathbb N^d$, which is a severe restriction and in general much harder to obtain.

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