Open sets of Axiom A flows with exponentially mixing attractors. (English) Zbl 1359.37066
Consider an Axiom A flow \(X^t\) on a Riemannian manifold with dimension bigger than two, a basic set \(\Lambda\) is an invariant, closed, topologically transitive, locally maximal hyperbolic set. If there is an open neighborhood \(U\) of \(\Lambda\) such that \(\Lambda=\cap_{t\geq0}X^t(U)\), then \(\Lambda\) is an attractor. An invariant measure is exponential mixing if the upper bound of the absolute value of the correlation function \(\rho_{\phi,\psi}(t)=\int_{\Lambda}\phi\circ X^t\cdot\psi d\mu-\int_{\Lambda}\phi d\mu\int_{\Lambda}\psi d\mu\) is controlled by an exponential function \(Ce^{-\gamma t}\), where \(\gamma>0\) is a constant, \(\phi\) and \(\psi\) are Hölder functions with the same exponent, \(C\) is constant depending on the Hölder norms of \(\phi\) and \(\psi\).
This paper concerns a problem of dynamical systems with exponential mixing rates: is this property robust (systems in a small open neighborhood also have the same property) or not? This is an open problem even for Anosov flows and Axiom A flows.
The authors construct a \(C^1\) open subset of \(C^3\) vector fields on any Riemannian manifold with dimension bigger than two, such that the flow for the vector field taken from this open set is Axiom A and mixes exponentially on a non-trivial attractor, where the invariant measure is the unique Sinai-Ruelle-Bowen measure (Theorem A).
This construction is based on another interesting result (Theorem B), where the authors study the exponential mixing rates of \(C^2\) Axiom A flows, and the critical assumptions are that the stable foliation is \(C^2\), and stable and unstable foliations are not jointly integrable.
The idea of the proof of Theorem B is the application of the Poincaré section and quotient along stable manifolds, such that one only needs to consider a suspension semiflow over an expanding Markov map, and apply results in [A. Avila et al., Publ. Math., Inst. Hautes Étud. Sci. 104, 143–211 (2006; Zbl 1263.37051)] to obtain the exponential mixing rates. The Markov partition is based on R. Bowen’s classical work [Am. J. Math. 95, 429–460 (1973; Zbl 0282.58009)], and the return time function is constant along the local stable manifolds.
The assumption of the \(C^2\)-stable foliations might be reduced to \(C^{1+\alpha}\) with some elegant arguments in future.
This paper concerns a problem of dynamical systems with exponential mixing rates: is this property robust (systems in a small open neighborhood also have the same property) or not? This is an open problem even for Anosov flows and Axiom A flows.
The authors construct a \(C^1\) open subset of \(C^3\) vector fields on any Riemannian manifold with dimension bigger than two, such that the flow for the vector field taken from this open set is Axiom A and mixes exponentially on a non-trivial attractor, where the invariant measure is the unique Sinai-Ruelle-Bowen measure (Theorem A).
This construction is based on another interesting result (Theorem B), where the authors study the exponential mixing rates of \(C^2\) Axiom A flows, and the critical assumptions are that the stable foliation is \(C^2\), and stable and unstable foliations are not jointly integrable.
The idea of the proof of Theorem B is the application of the Poincaré section and quotient along stable manifolds, such that one only needs to consider a suspension semiflow over an expanding Markov map, and apply results in [A. Avila et al., Publ. Math., Inst. Hautes Étud. Sci. 104, 143–211 (2006; Zbl 1263.37051)] to obtain the exponential mixing rates. The Markov partition is based on R. Bowen’s classical work [Am. J. Math. 95, 429–460 (1973; Zbl 0282.58009)], and the return time function is constant along the local stable manifolds.
The assumption of the \(C^2\)-stable foliations might be reduced to \(C^{1+\alpha}\) with some elegant arguments in future.
Reviewer: Xu Zhang (Weihai)
MSC:
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37C10 | Dynamics induced by flows and semiflows |
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