I've been trying to solve the following problem
" Considering a measurable dinamical system $(X, \mathcal{B}, \mu, \mathcal{T})$ where $\mathcal{T}$ is an action of a semigroup $G = N^d$ on $X$ for $d \geq 2$. For $g^* \in G \backslash \{0\}$ define the unidimensional subaction of the group $G' = N_0$ by $T_g^* =\{ T_{n \cdot g^*}\}_{n \in G'}$. Show that if $h_\mu(T_g^*) < \infty$, the $h_\mu(\mathcal{T}) = 0$."
I've tried different ways of bounding the entropy of the system by partitions in the direction defined by $g^*$ but I either come up with a calculation that is too hard to manipulate or with a bound that does not allow to conclude as desired.
Any help is apreciated