Extreme growth rate of periodic orbits for equivalent differentiable flows

Abstract

We prove that the extreme growth rate of periodic orbits of differentiable flows is preserved by Lipschitz equivalence when all the fixed points are hyperbolic and thus it is an invariant in an open dense subset X h r of the set of C r ( 1 ≤ r ≤ ∞ ) flows. By contrast, for any 1 ≤ r ≤ ∞ , there exist two infinite dimensional connected subsets of equivalent C r flows disjoint from X h r such that the growth rate of periodic orbits attains infinity and zero respectively.