Let be a continuous map. CAO Yongluo proved that if is continous and for all , then

for EVERY , there exists such that the Birkhoff average ;

Moreover there exists a uniform such that for every and .

Now let be a family of continuous subadditive potential with respect to . By subadditive ergodic theorem we know that the limit exists on a set of full probability.

CAO: If on a set of full probability, then there exists a uniform and such that

for every .

Proof. For each we have .

So there exists such that .

By the continuity of , there exists an open set such that

for all .

Since is compact, finite open sets, say , cover .

Now let and . QED.

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Collecting terms together, we see that there exists and such that

for every and .

By the subadditive assumption, we see

for each . So

————-. (indepedent of .)

Then we see for another large, for all and .

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For example we let be a local diffeomorphism and . Then is continuous and . Cao proved that if the Lyapunov exponent on a full probability set, then for every , and hence is uniformly expanding.

Another application is that if is a continuous splitting such that on a full probability set, then is uniformly contracting.

In a sequent paper they constrcuted a homoclinic class (contains a tangency, hence nonhyperbolic) such that all is uniformly hyperbolic: there exists such that on a set of full probability. In particular all periodic points are uniformly hyperbolic after finishing their periods.