Tag Archives: Axiom A flow

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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Mixing but non-exponentially mixing Axiom A flows: Ruelle’s example

Bowen extended Sinai’s construction of Markov partition to Axiom A diffeomorphism, and proved that the mixing rate of every Gibbs measure \mu with respect to a Holder potential over a mixing basic set is always exponential, that is,
for all smooth functions \phi,\psi on M, the correlations \rho(t)=\int \phi\cdot\psi\circ \sigma^n d\mu-\mu(\phi)\cdot\mu(\psi)\to 0 exponentially.
However the situation is quite different for Axiom A flows. Ruelle gave the first class of examples: mixing but non-exponentially mixing Axiom A flows.

Ruelle first constructed a symbolic example and then embedded it into an Axiom A flow. Let \Omega=\{0,1\}^{\mathbb{Z}} and \sigma be the shift on \Omega. Pick two positive numbers \lambda_0<\lambda_1 such that \frac{\lambda_0}{\lambda_0} is irrational. Define a ceiling function \tau:\omega\in\Omega\mapsto \lambda(\omega_0). Then let (\Omega_\tau,\sigma_t) be the suspension flow over (\Omega,\sigma) with respect to \tau. It is well known that there is a one to one correspondence between the \sigma_t-invariant measures \nu and \sigma-invariant measures \mu: d\nu=\frac{dt\times d\mu}{\mu(\tau)}. Ruelle examined the measure of maximal entropy, \mu (corresponding the zero potential) and showed that the corresponding measure \nu does not mix exponentially under \sigma_t.

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