On Bowen entropy inequality for flows.

We study the set of generic points of continuous flows on compact metric spaces. We define an entropy for noncompact sets for flows and establish in this setting the celebrated result by Bowen in [3]: we prove that for any invariant measure $\mu$ of a continuous flow $\Phi$ it holds $ h(\Phi,G_\mu(\Phi))\leq h_\mu(\Phi)$, where $G_\mu(\Phi)$ is the set of generic points of $\Phi$ respect to $\mu$. Moreover, the equality is true for ergodic measures. We also derive some immediate consequences from this result about the saturated and irregular points of flows, which can be seen as an extension of the results of Pfister and Sullivan in [13], and Thompson in [15,16].