Extreme growth rate of periodic orbits for equivalent differentiable flows (Q2633329)

The authors investigate differentiable flows on a compact Riemannian manifold and examine, in particular, the question of what quantities are preserved when two flows are equivalent. (Two flows on a compact topological space are equivalent if there is a homeomorphism that takes each orbit of one onto an orbit of the other while preserving the time orientation.) They note that for flows without fixed points, the extreme values of 0 and $\infty$ of the topological entropy are preserved by equivalence. With fixed points, however, the topological entropy is not necessarily preserved under equivalence. \par The work is motivated by the common belief that ``nonzero entropy'' coexists with ``nonzero exponential growth rate of periodic orbits'' for a broad collection of systems. In this paper they focus on the extreme growth rate of periodic orbits for differentiable flows with hyperbolic fixed points. Their measure for the exponential growth rate of periodic orbits for a flow $\phi$ is \[{\mathrm{EP}}(\phi) = \limsup_{t\rightarrow\infty} \frac{1}{t} \log \widehat P_t(\phi), \] where \(\widehat P_t (\phi)\) is the maximum of 1 and the number of isolated orbits $\phi$ such that for some \(s \in [t, t+1]\), \(\phi(x,0)= \phi(x,s)\) and \(\phi(x,0) \neq \phi(x, \tau)\) when \(\tau \in (0, s)\). When $\phi$ is an Anosov flow, the topological entropy is equal to EP$(\phi)$. This does not hold in the general case. \par The main result is as follows: if $\phi$ and $\psi$ are two differentiable flows on a compact Riemannian manifold that are equivalent by a Lipschitz homeomorphism, and if all the fixed points of $\phi$ and $\psi$ are hyperbolic, then EP($\phi$) = 0 if and only if EP($\psi$) = 0 and EP($\phi$) = $\infty$ if and only if EP($\psi$) = $\infty$. Furthermore, since the set of $C^r$ flows whose fixed points are all hyperbolic for \(1 \le r \le \infty\) is an open and dense subset of all $C^r$ flows, the extreme values of 0 and $\infty$ are preserved by equivalence in the latter subset.