## 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.

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.

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.
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 $\sigma_t$.