Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part (Q2884418)

The existence and finiteness of measures of maximal entropy is an important topic in smooth ergodic theory and is well known for classical hyperbolic systems. Moreover the system is said to be \textit{intrinsically ergodic} if there exists a unique measure of maximal entropy. As pointed out after Theorem 2, the paper under review is complementary to the work [\textit{F. Rodriguez Hertz} et al., Ergodic Theory Dyn. Syst. 32, No. 2, 825--839 (2012; Zbl 1257.37024)].NEWLINENEWLINEThe author proves the intrinsic ergodicity for a class of partially hyperbolic diffeomorphisms on \(\mathbb{T}^n\), and gives several characterizations of this measure. More precisely, let \(f:\mathbb{T}^n\to\mathbb{T}^n\) be a \(C^1\) absolutely partially hyperbolic diffeomorphism homotopic to a hyperbolic linear automorphism \(A\), such that \(\dim E^c=1\), and both stable and unstable foliations are quasi-isometric. Then \(f\) is intrinsically ergodic, that is, there exists a unique measure of maximal entropy for \(f\), say \(\mu\). Moreover \((f,\mu)\) and \((A,m)\) are isomorphic. Here \(m\) denotes the Lebesgue measure on \(\mathbb{T}^n\).NEWLINENEWLINEIn fact, the author gives a neat statement for 3-dimensional case: Let \(f:\mathbb{T}^3\to\mathbb{T}^3\) be an absolutely partially hyperbolic diffeomorphism homotopic to a hyperbolic linear automorphism. Then the same conclusions hold. In particular \(f\) is intrinsically ergodic.NEWLINENEWLINETwo ingredients for this 3-dimensional case are \textit{M. Brin} et al.'s result on quasi-isometric strong foliations [J. Mod. Dyn. 3, No. 1, 1--11 (2009; Zbl 1190.37026 )] and \textit{A. Hammerlindl}'s work on quasi-isometric center foliation [``Leaf conjugacies on the torus'', Erg. Th. Dyn. Sys. (to appear) \url{doi:10.1017/etds.2012.171}].NEWLINENEWLINEThe author also proves an interesting estimate for the center Lyapunov exponent of \(\mu\). Namely, let \(f:\mathbb{T}^3\to\mathbb{T}^3\) be a \(C^{1+\alpha}\) absolutely partially hyperbolic diffeomorphism homotopic to a hyperbolic linear automorphism \(A\) and \(\mu\) be the measure of maximal entropy given above. If \(\lambda^c(A)>0\), then \(\lambda^c(\mu)\geq \lambda^c(A)\). Similarly, if \(\lambda^c(A)<0\), then \(\lambda^c(\mu)\leq \lambda^c(A)\).NEWLINENEWLINEHis proof is based on a Pesin-Ruelle type inequality proved by \textit{Y. Hua} et al. in [Ergodic Theory Dyn. Syst. 28, No. 3, 843--862. (2008; Zbl 1143.37023)], namely \(h_\nu(f)\leq\chi_u(f)+\sum_{\lambda^c_i(\nu)>0}\lambda^c_i(\nu)\). As pointed out in Question 5.3, it would be interesting to know if the estimates of the center Lyapunov exponent holds for higher dimensions.NEWLINENEWLINENow let's move back to the \(n\)-dimensional case and assume \(\lambda^c(A)>0\). The author proves that the support of \(\mu\) is saturated by the stable leaves and is the unique minimal set of the stable foliation of \(f\). The author also raises the following questions: is \(\mu\) fully supported? Is \(f\) topologically transitive?