## 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.

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.

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 $\Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x)$.
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. $x$.