Ergodic properties of equilibrium measures for smooth three dimensional flows. (English) Zbl 1366.37087
The authors consider the problem of understanding when a smooth flow is a Bernoulli automorphism. This problem was extensively studied by D. Ornstein and B. Weiss [Ergodic Theory Dyn. Syst. 18, No. 2, 441–456 (1998; Zbl 0915.58076)] and A. Katok and K. Burns [ibid. 14, No. 4, 757–785 (1994; Zbl 0816.58029)] among others. The purpose of this article is to study the ergodic structure of measures of maximal entropy for flows in three dimensional manifolds and with positive entropy.
One the main results (Theorem 1.1) states that if \(\mathbf T=\{T_t:M\to M\}\) is a flow on a compact three dimensional \(C^\infty\) manifold \(M\), which is generated by a \(C^{1+\varepsilon}\) vector field \(X\) on \(M\), then any equilibrium measure \(\mu\) for a Hölder continuous potential has at most countable many ergodic components \(\mu_n\) with positive entropy and such that \(\mathbf T\) is Bernoulli up to a period with respect to each measure. This means that the flow is Bernoulli or isomorphic to a product of a Bernoulli flow and a rotational flow.
One the main results (Theorem 1.1) states that if \(\mathbf T=\{T_t:M\to M\}\) is a flow on a compact three dimensional \(C^\infty\) manifold \(M\), which is generated by a \(C^{1+\varepsilon}\) vector field \(X\) on \(M\), then any equilibrium measure \(\mu\) for a Hölder continuous potential has at most countable many ergodic components \(\mu_n\) with positive entropy and such that \(\mathbf T\) is Bernoulli up to a period with respect to each measure. This means that the flow is Bernoulli or isomorphic to a product of a Bernoulli flow and a rotational flow.
Reviewer: Alejandro Mesón (La Plata)
MSC:
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37B10 | Symbolic dynamics |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37C10 | Dynamics induced by flows and semiflows |
| 37C35 | Orbit growth in dynamical systems |
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