nonuniform conditions and uniform conclusions

Let f:X\to X be a continuous map. CAO Yongluo proved that if \psi\in C(X,\mathbb{R}) is continous and \mu(\psi) < \lambda for all \mu\in\mathcal{M}(f), then
for EVERY x\in X, there exists n_x\ge 1 such that the Birkhoff average \psi_f(x,n_x) < \lambda;
Moreover there exists a uniform N\ge1 such that \psi_f(x,n) < \lambda for every x\in X and n\ge N.

Now let \phi_n: X\to \mathbb{R} be a family of continuous subadditive potential with respect to (X,f). By subadditive ergodic theorem we know that the limit \phi_f(x)=\lim_{n\to\infty} \frac{\phi_n(x)}{n} exists on a set of full probability.

CAO: If \phi_f(x) < 0 on a set of full probability, then there exists a uniform L\ge 1 and \lambda < 0 such that
\int \frac{\phi_N}{N} d\mu < \lambda for every \mu\in \mathcal{M}(f).

Proof. For each \mu\in \mathcal{M}(f) we have \inf_{n\ge1} \int \frac{\phi_n}{n}d\mu =  \int \phi_f d\mu < 0.
So there exists n_\mu \ge 1 such that \int \frac{\phi_{n_\mu}}{n_\mu} d\mu\le \mu(\phi_f)/2 < 0.

By the continuity of \phi_n, there exists an open set \mathcal{U}_\mu \ni \mu such that
\int \frac{\phi_{n_\mu}}{n_\mu} d\nu\le \mu(\phi_f)/3 < 0 for all \nu \in \mathcal{U}_\mu.

Since \mathcal{M}(f) is compact, finite open sets, say \mathcal{U}_1,\cdots,\mathcal{U}_k, cover \mathcal{M}(f).

Now let \lambda= \min(\mu_i(\phi_f)/3:1\le i\le k) and L=\prod_i n_i. QED.

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Collecting terms together, we see that there exists N\ge L\ge1 and \lambda < 0 such that

\phi_L(x,n) < \lambda for every x\in X and n\ge N.

By the subadditive assumption, we see

\phi_{kL}(x)\le \phi_j(x)+\sum_i^{k-1} \phi_L(f^{Li+j}x)+\phi_{j'}(f^{kL-L+j}x) for each 0\le j\le L. So

\phi_{kL}(x)\le \sum_j^L\phi_j(x)/L +\sum_j^L\sum_i^{k-1} \phi_L(f^{Li+j}x)/L+\sum_j^L\phi_{j'}(f^{kL-L+j}x)/L

————-=*+\sum_n^{kL-L}\phi_L(f^nx)/L+*\le 2C+(k-1)L\lambda. (indepedent of x\in X.)

Then we see for another K large, \frac{\phi_n(x)}{n}\le \lambda/2 for all n\ge K and x\in X.

——————————————————————

For example we let f:M\to M be a local diffeomorphism and \phi_n(x)=\log \|Df^{-n}:T_{f^nx}M\to T_xM\|. Then \phi_n is continuous and \phi_{m+n}(x) \le\phi_n(x)+\phi_m(f^nx). Cao proved that if the Lyapunov exponent \lambda(f,x) =-\phi_f(x) > 0 on a full probability set, then m(D_xf^n)\ge e^{n\delta} for every x\in M, n\ge N and hence f is uniformly expanding.

Another application is that if TM=E\oplus F is a continuous splitting such that \lambda_f(x,E)   < 0 on a full probability set, then Df|_E is uniformly contracting.

In a sequent paper they constrcuted a homoclinic class \Lambda (contains a tangency, hence nonhyperbolic) such that all \mu\in \mathcal{M}(f,\Lambda) is uniformly hyperbolic: there exists c  >0 such that |\lambda^i(x,f)|\ge c on a set of full probability. In particular all periodic points are uniformly hyperbolic after finishing their periods.

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