I asked a question about entropy here
but I think it might me a little too esoteric so I'm going to try and rephrase it. Let $Y$ be a compact group, and suppose that $\beta$ is a finite partition of $Y$, meaning that $Y$ is the union of all elements of $\beta$ and the elements of $\beta$ are pairwise disjoint. For two partitions $\alpha$ and $\beta$ define $\beta \vee \alpha=\{A_i \cap B_j : A_i \in \alpha, B_j \in \beta\}$. Let $g_i \in Y$ and consider $\beta_1^n=g_1 \beta \vee g_1g_2 \beta \vee \dots \vee g_1g_2 \dots g_n \beta$. Are there constants $K$ and $m$ such that the number of elements of $\beta_1^n$ is bounded by $(Kn)^m$? The end goal is to get some bound $M_n$ on the size of $\beta_1^n$ such that $log(M_n)/n \to 0$