Relation between entropy of one-parameter group and single elements of this group

Question

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.

Answer 1

A small precision :

If $\mu$ is an ergodic measure for the action of the one-parameter group $H=(h_t)_{t\in\mathbb{R}}$, then for almost every $t\in \mathbb{ R}$, the single element $h_t$ is ergodic.

Moreover, if $\mu$ is mixing (it is the case for example if $\mu$ is the Liouville measure or any other Gibbs measure, in your favorite case), then for all $t\in\mathbb{R}$, the single element $h_t$ is mixing.

And as said in the previous answer, the entropy of $H$ is the entropy of $h_1$ or the entropy of $h_t$ divided by $|t|$.

Answer 2

The entropy of a flow $H=(h_t)$ is defined as the entropy of the time 1 transformation $h_1$. For any other $t$ the entropy of $h_t$ is $|t|$ times the entropy of $h_1$ (for example, see the book by Cornfeld-Fomin-Sinai).