Let $\Phi=(\phi_t)_{t\in \mathbb{R}}$ be a continuous flow on a compact metric space $X$. Let $\mu$ be a $\phi_1$-invariant measure. Then it is not hard to verify tht $\int_{0}^{1} \phi_t\mu dt$ is $\Phi$-invariant measure. I would like to whether it is possible to compare the value of entropies $h(\mu)$ and $h(\int_{0}^{1} \phi_t\mu dt)$. Are they equal? Thanks.
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