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We study the Bowen topological entropy of generic and irregular points forcertain dynamical systems. We define the topological entropy of noncompact setsfor flows, analogous to Bowen's definition. We show that this entropy coincideswith the Bowen topological entropy of the time-1 map on any set. We also show aBowen's inequality for flows; namely, that the metric entropy with respect toevery invariant measure for a continuous flow is an upper bound for thetopological entropy of the set of generic points with respect to the samemeasure, and the equality is always true if the measure is ergodic. We proposea definition of almost specification property for flows and prove that acontinuous flow has the almost specification property if the time-1 mapsatisfies this property. Using Bowen's inequality for flows, we show that everycontinuous flow with the almost specification property is saturated, extendinga result of Meson and Vericat in [22]. Under the same hypotheses, we extend aresult of Thompson on the entropy of irregular points in [30].

On Bowen's entropy inequality and almost specification for flows