My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ergodic theory material that is available to me. So, I would be grateful for any answer or reference.

Here is the question: Given a measure space $(X,\mu)$ and a one-parameter group $H=\{h_t: t\in\mathbb{R}\}$ acting on $X$ (in any nice way that might be necessary). The example I have in mind is $X=\Gamma\backslash SL_2(\mathbb{R})$ for some lattice $\Gamma$ in $SL_2(\mathbb{R})$ and $H$ the diagonal subgroup $diag(e^t, e^{-t})$. Pick an element $h\in H$, $h\not= id$. I want to compare the entropy of the ergodic components of $\mu$ with respect to $H$ to those with respect to $h$. Are the following statements equivalent?

1) For almost all $x\in X$, the entropy of the $H$-ergodic component is positive. 2) For almost all $x\in X$, the entropy of the $h$-ergodic component is positive.