Closed orbits and homology for \(C^2\)-flows (Q1576847)

Let \(M\) be a compact three-manifold and \(\varphi_t: M\to M\) be a \(C^2\) flow on it. Let \(\mu\) be any \(\varphi\)-invariant probability measure. By \(E_{\text{erg}}\), \(H^1(M,R)\) are denoted a set of ergodic measures and first cohomology group, respectively. Given an open set \(W\subset H_1(M, R)\) one can define \[ \beta_W(\varphi)= \sup\{h(\varphi, \mu)\mid\mu\in E_{\text{erg}}\text{ such that }\Phi_\mu\subset W\}, \] where \(\Phi_\mu: H^1(M, R)\to R\) is defined standard way, that is \(\Phi_\mu([\omega])= \int\omega(X) d\mu\). Here by \(X\) and \([\omega]\) are denoted associated to \(\varphi_t\) the vector field and corresponding element in the de Rham cohomology, respectively. Let \(\gamma\) denote a closed orbit for the flow of least period \(\lambda(\gamma)\), \([\gamma]\in H_1(M,R)\) the homology class in which \(\gamma\) lies and \[ \pi_W(T)= \text{Card}\Biggl\{\gamma\mid \lambda(\gamma)\leq T\text{ and }{1\over\lambda(\gamma)} [\gamma]\in W\Biggr\} \] for \(T> 0\). The main result of the author is the following. Theorem: Let \(\varphi_t\) be a flow mentioned above. If \(W\subset H_1(M,R)\) is an open set then \[ \lim_{T\to+\infty} \sup{1\over T}\log \pi_W(T)\geq \beta_W(T). \] The proof of the theorem is based on the variational principle for entropy. In the case \(W= H_1(M,R)\) we obtain \(\beta_{H^1(M,R)}(\varphi)= h(\varphi)\). Hence, the author generalizes a well-known result of A. Katok. Here by \(h(\varphi)\) is denoted the topological entropy of the (time-one) flow. The \(C^2\) assumption for the flow \(\varphi_t: M\to M\) can be weakened to \(C^{1,\delta}\) for any \(\delta> 0\).