Entropy and Periodic Orbits for Equivalent Smooth Flows

Given any K > 0, we construct two equivalent C2 flows, one of which has positive topological entropy larger than K and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and super-exponential growth of periodic orbits. Moreover we establish a C∞ flow on S2 with super-exponential growth of periodic orbits, which is also equivalent to another flow with zero exponential growth of periodic orbits. On the other hand, any two dimensional flow has only zero topological entropy.