Hyperbolic measures with transverse intersections of stable and unstable manifolds
Let \(f:M\rightarrow M\) be a \(C^{1+\alpha}\) diffeomorphism of a compact smooth Riemannian manifold \(M\) preserving a hyperbolic Borel probability measure \(\mu\). Consider the generic set of the measure \(\mu\) defined by \[ G_\mu = \{x \in M: \lim_{n\to +\infty} \delta_n(x) = \mu \} \] where \(\delta_n(x)=\frac 1n \sum_{i=0}^{n-1} \delta(f^i(x))\) and \(\delta(y)\) denotes the Dirac measure at \(y\). Suppose that \(\mu\) satisfies the almost transversality condition, i.e., for \(\mu\otimes\mu\)-almost every pair \((x,y)\in M\times M\) there exist integers \(p,q\in \mathbb{Z}\) and a point \(z\in W^u(f^p(x))\cap W^s(f^q(y))\) such that \[ T_zW^u(f^p(x))\oplus T_zW^s(f^q(y))=T_zM, \] where \(W^u(\cdot)\) and \(W^s(\cdot)\) denote the unstable and stable manifold respectively. This paper proves the following main results: (1) \(G_\mu\) is not empty and \(h(f,G_{\mu})=h_{\mu}(f)\), where \(h(f,G_{\mu})\) is the topological entropy of \(G_\mu\) and \(h_\mu(f)\) is the metric entropy; (2) the Hausdorff dimension of the generic set satisfies that \[ \dim_H G_{\mu}\geq d^s + \frac{h_\mu(f)}{\int \chi_1(x)d\mu}, \] where \(d^s\) is the dimension of the stable manifold and \(\chi_1(x)=\lim\limits_{n\to\infty} \frac 1 n \log \|D_xf^n\|\) is the maximal Lyapunov exponent; (3) for any open neighborhood \(\mathcal{G}\) of \(\mu\) in the space of all probability measures on \(M\), the following holds \[ \liminf\limits_{n\to \infty} \frac 1n \log m(\{x\in M: \delta_n(x)\in \mathcal{G}\}) \geq h_{\mu}(f)-\int \chi^+(x) d\mu, \] where \(\chi^+(x) = \lim\limits_{ n\to \infty}\frac 1n \log |\mathrm{det}(D_xf^n|T_xW^u(x))|\).
scientific article; zbMATH DE number 6157262