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 $x\in X$.

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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.