Tag Archives: unstable manifold

Basic set of a smooth flow: Bowen’s trichotomy

This is a note taken from Bowen, Periodic Orbits for Hyperbolic Flows, American Journal of Mathematics (1972), 1–30.

We start with Anosov dichotomy:

Let f:M\to M be a transitive Anosov flow. Then

1. either it is mixing: then strong stable and strong unstable manifold everywhere dense on M
2. or it is a suspension: choose the closure of a non-dense stable manifold as a cross-section and the induced roof-function is constant.

Anosov proved this in volume-preserving case and Plante proved it for the general case (Anosov flows, 1972).

Let M be a closed manifold and \phi_t:M\to M be a C^1 flow. Let \Omega be a closed, invariant subset without fixed points, that is, the vector field of \phi does not admits zeros on \Omega. Then \Omega is said to be a basic set of \phi if
(\Omega,\phi_t) is (topologically) transitive and hyperbolic,
– close orbits are dense in \Omega
(\Omega,\phi_t) is isolated: \Omega=\bigcap_{\mathbb{R}}\phi_t(U) for some open neighborhood U of \Omega.
The last one is equivalent to the local product structure.

Theorem 3.2 in [B]. There are three mutually exclusive types:
1. \Omega consists of a single closed orbit of \phi;
2. the strong stable manifold W^s(p) is dense in \Omega for for each p\in\Omega;
3. (\Omega,\phi_t) is the constant suspension of a Axiom A homeomorphism.

As remarked by Bowen, this is first proved by Anosov for volume-preserving Anosov flow, by Plante for general Anosov flow. We should view the following as a proof for Anosov flow case for first reading, and then take the induced topology on \Omega for basic sets.

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4. Ledrappier-Young’s criterion for measures with Absolutely Continuous Conditional measures on Unstable manifolds. Let f:M\to M be a C^2 diffeomorphism and \mu be an invariant probability measure. For \mu-a.e. x, if there exist negative Lyapunov exponent(s) at x, then the set of points with exponentially approximating future of x is a C^2-submanifold, W^u(x). Then \mu is said to have ACCU if for each measurable partition \xi with \xi(x)\subset W^u(x) and contains an unstable plaque for \mu-a.e. x, the conditional measure \mu_{\xi(x)}\ll m_{W^u(x)} for \mu-a.e. x.

Theorem. \mu has ACCU if and only if h_\mu(f)=\Lambda^+(\mu), where \Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x).

Definition. Such a measure is called the SRB measure.

Moreover, the density \frac{d\mu_{\xi(x)}}{dm_{W^u(x)}} is strictly positive and C^1 on \xi(x) for \mu-a.e. x.

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