A rescaled metric entropy of C1-vector field with singularities

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For a C1-vector field on a closed Riemannian manifold, we add the flow speed control to define Bowen ball, and the new Bowen balls are given after considering reparametrization. That use the idea of rescaling the size of the neighborhoods of a regular point. Then we give new definitions of metric-entropy by using the minimum number of these new types of Bowen balls covering a set and the measure of this set is more than 1−δ. We prove that these new definitions are equivalent, and we call them rescaled metric entropy. Then we prove that for a C1-vector field, the rescaled metric entropy is equal to the metric entropy of the time-1 map of the flow providing that ln⁡‖X(x)‖ is integrable.