Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part.(English)Zbl 1258.37033
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 beintrinsically 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 [F. Rodriguez Hertz et al., Ergodic Theory Dyn. Syst. 32, No. 2, 825–839 (2012;Zbl 1257.37024)].
The 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 \(\mathbb{T}^n\).
In 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.
Two ingredients for this 3-dimensional case areM. Brin et al.’s result on quasi-isometric strong foliations [J. Mod. Dyn. 3, No. 1, 1–11 (2009;Zbl 1190.37026 )] andA. Hammerlindl’s work on quasi-isometric center foliation [“Leaf conjugacies on the torus”, Erg. Th. Dyn. Sys. (to appear)doi:10.1017/etds.2012.171].
The 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 \(\lambda^c(\mu)\leq \lambda^c(A)\).
His proof is based on a Pesin-Ruelle type inequality proved byY. 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.
Now 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?
The 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 \(\mathbb{T}^n\).
In 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.
Two ingredients for this 3-dimensional case areM. Brin et al.’s result on quasi-isometric strong foliations [J. Mod. Dyn. 3, No. 1, 1–11 (2009;Zbl 1190.37026 )] andA. Hammerlindl’s work on quasi-isometric center foliation [“Leaf conjugacies on the torus”, Erg. Th. Dyn. Sys. (to appear)doi:10.1017/etds.2012.171].
The 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 \(\lambda^c(\mu)\leq \lambda^c(A)\).
His proof is based on a Pesin-Ruelle type inequality proved byY. 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.
Now 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?
Reviewer: Pengfei Zhang (Amherst)
MSC:
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
Keywords:
intrinsic ergodicity;partial hyperbolicity;meaure of maximal entropy;quasi-isometric foliations;central Lyapunov exponentCite
Full Text:DOI
References:
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