Tag Archives: stable saturate

Stable and unstable saturation

Let (X,d) be a compact metric space, f:X\to X be a homeomorphism preserving some probability measure \mu. The stable set W^s(x) is defined by the formula
W^s(x)=\{y\in X: d(f^nx,f^ny)\to 0\text{ as }n\to+\infty\}. For convenience denote the collection of null sets as \mathcal{N}=\{N\subset X:\exists \tilde{N}\supset N, \mu(\tilde{N})=0\}.

[Lemma 6.3.2. Brin M., Stuck G. Introduction to dynamical systems]
Let \phi:X\to [0,1] be an f-invariant measurable function. Then there exists a subset N\in\mathcal{N} such that for each x\in X\backslash N, \phi is constant on W^s(x)\backslash N.

Proof. Pick a sequence of continuous function \phi_k, k\ge1 on X such that
\phi_k\to \phi both L^1(\mu) and \mu-a.e..

By Birkhoff Ergodic Theorem, the limit \phi_k^+(x)=\lim_{n\to+\infty}\frac{1}{n}\sum_{i=0}^{n-1}\phi_k(f^ix) exists for \mu-a.e.\; x. So there exists N_k\in\mathcal{N} such that the limit exists for each x\in X\backslash N_k. Pick such a point x. Then for each y\in W^s(x), the limit also exists at y and has the same value as at x: \phi_k^+(y)=\phi_k^+(x). (both N_k,X\backslash N_k are stable-saturated)

By the invariance of \mu and \phi, we have
\int|\phi(y)-\frac{1}{n}\sum\limits_{i=0}^{n-1}\phi_k(f^iy)|d\mu(y)\le\frac{1}{n}\sum\limits_{i=0}^{n-1}\int|\phi(y)-\phi_k(f^iy)|d\mu(y)=\|\phi-\phi_k\|.

By Dominated Convergent Theorem, \|\phi-\phi_k^+\|\le\|\phi-\phi_k\|\to 0. Passing to a subsequence if necessary, we assume that \phi_k^+\to\phi,\:\mu-a.e.\:x. So there exists N_+\in\mathcal{N} such that for each x\in X\backslash N_+, the limit \lim_{k\to\infty}\phi_k^+(x) exists and equals to \phi(x).

Now let N=\bigcup_{k\ge1}N_k\cup N_+. Then N\in\mathcal{N} and for each point x\in X\backslash N, each point y\in W^s(x)\backslash N, \phi_k^+(y)=\phi_k^+(x) for each k\ge1 and
1. the limits exist (hence are equal): \lim_{k\to\infty}\phi_k^+(y)=\lim_{k\to\infty}\phi_k^+(x).
2. they equal to \phi respectively: \lim_{k\to\infty}\phi_k^+(x)=\phi(x), \lim_{k\to\infty}\phi_k^+(y)=\phi(y).

So we have \phi(y)=\phi(x) for each y\in W^s(x)\backslash N and x\in X\backslash N.

A more interesting application:
Corollary 1. Let (X,f), \mu\in\mathcal{M}(f) and E\subset X be an invariant subset. Then there exists a null set N\in\mathcal{N}_\mu such that for each x\in E\backslash N, W^s(x)\backslash N \subset E. (equivalently, W^s(x)\backslash E \subset N)

Proof. The characteristic function \chi_E is invariant and measurable. Let N be given by Lemma. Then if x\in E\backslash N, we have \chi_E(y)=\chi_E(x)=1 for each y\in W^s(x)\backslash N. So W^s(x)\backslash N\subset E.

Similarly result holds for unstable sets. The results can be imporved if the stable/unstable foliations are absolutely continuous. This is the case for nonuniformly hyperbolic invariant sets.

Corollary 2. Let f\in \mathrm{Diff}^r_\mu(M) for some r>1 as above. Let \Lambda\subset M be an invariant nonuniform hyperbolic subset. Then there exists a null set N\in\mathcal{N}_\mu such that for each x\in \Lambda\backslash N, \mu_{W^s(x)}(W^s(x)\backslash \Lambda)=0 and \mu_{W^u(x)}(W^u(x)\backslash \Lambda)=0.

Equivalently, we have for \mu-a.e. x\in\Lambda, W^s(x)\subset \Lambda\; \mathrm{mod}\mu_{W^s(x)} and W^u(x)\subset \Lambda\; \mathrm{mod}\mu_{W^u(x)}.