Variational Principle for Fixed-point Free Flow

Let (X, ϕ) be a compact flow without fixed points. We define the packing topological entropy h ^P_top (ϕ, K) on subsets of X through considering all the possible reparametrizations of time. For fixed-point free flows, we prove the following result: for any non-empty compact subset K of X

$$h_{\text{top}}^{P}(\phi, K) = \sup \left\{\overline{h}_{\mu}(\phi) : \mu(K) = 1,\ \mu\, \text{is}\, \text{a}\, \text{Borel}\, \text{probability}\, \text{measure}\, \text{on}\, X\right\},$$

where \(\overline{h}_{\mu}(\phi)\) denotes the upper local entropy for a Borel probability measure μ on X.