A global volume lemma and applications (Q1182833)

Let \(M\) be a compact Riemann manifold and let \(f^ t: M\to M\) be a \(C^ 2\) dynamical system satisfying Axiom A and strong transversality condition, where \(t\in\mathbb{R}\) or \(t\in\mathbb{Z}\). Denote by \({\mathcal I}_ t(x)\) the Jacobian of the linear map \(Df^ t\) restricted to the unstable subbundle and \(\Phi^ u(x)=-(d{\mathcal I}_ t(x)/dt|_{t=0}\) in the continuous case and \(\Phi^ u(x)=\log{\mathcal I}_ t(x)\) in the discrete time case. Let \(\Phi\) be an admissible Hölder continuous extension of \(\Phi^ u\) to all \(M\) and let \(B_ x(\varepsilon,t)=\{y:\text{dist}(f^ sx,f^ ty)\leq \varepsilon\) for all \(s\in[0,t]\}\). The authors prove that for small \(\varepsilon > 0\) there exists a constant \(C_ \varepsilon>0\) depending on \(\Phi\) such that for any \(x\in M\) and \(t\geq 0\), \[ C^{-1}_ \varepsilon\leq\text{vol}(B_ x(\varepsilon,t))\exp(-S^ \Phi_ t(x))\leq C_ \varepsilon, \] where vol denotes the Riemannian volume and \(S^ \Phi_ t(x)=\sum^{t- 1}_{n=1}\Phi(f^ nx)\) in the discrete time case and \(S^ \Phi_ t(x)=\int^ t_ 0\Phi(f^ ux)du\) in the continuous case. This theorem generalizes the result of \textit{R. Bowen} and \textit{D. Ruelle} [Invent. Math. 29, No. 3, 181-202 (1975; Zbl 0311.58010)].