A simple fact: let be a homeomorphism preserving the measure-class of , be the Jacobian of at . Then for any sequence with we have for -a.e. . For example with or with . This is a direct corollary of Borel-Cantelli property:

Consider the set . It is easy to see and hence . So for all

Notes from papers by Jaksic, Pillet, Rey-Bellet, Ruelle and Young.

4. Let be two probability measures on and be its density. Then . Moreover is well-defined with respect to and , too.

The relative entropy can be defined as when , otherwise.

Note that . So the relative entropy is nonnegative.

Convexity: assume and , then and .

3. Let be a homeomorphism, and be a continuous function on . Then a probability on is said to be a Gibbs measure with respect to , if there exist with for all , such that for all .

Prop. Assume there exists some Gibbs measure for . Then for any continuous function , the moment generating function .

Proof. Let and . Then .

Assume is expansive and admits a unique equilibrium state, say . Then the pressure function is differentiable at and . Then according to Prop., for all . In particular we see for every -Gibbs measure .

Remark. The Gartner-Ellis Theorem actually says that for every -Gibbs measure , for -a.e. .

2. Let be a local homeomorphism preserving the measure-class of . Suppose there is an decomposition , and the branch inverse . Let be the Jacobian of , and be the induced transfer operator: . Then for any measurable function : .

Proof.

.

Moreover for any two functions , we have

.

An -abs. cts measure is -invariant if and only if .

Proof. .

Then we generalize the transfer operator . The moment generating function of with respect to an -invariant measure is . This is related to the transfer operator :

.

In particular, if has a spectral gap, then is analytic and leads to large deviation results of .

1. Let be a diffeomorphism, a pre-chosen smooth probability volume, the Jacobian of with respect to . The entropy production of is (that is, entropy is non-decreasing). Moreover, if and only if is -invariant (that is, an equilibrium state).

Ruelle introduced the entropy production for general -invariant measure as . Clearly , since is -invariant. Compare with .

It is observed in his paper that be Oseledec Multiplicative Ergodic Theorem, and hence independent of the choice of volumes.

Theorem 1.2. (1). Suppose , then . (He called such measures SRB there)

(2). Assume is hyperbolic SRB. Then if and only if is singular.

(3). if .

Proof. (1). by Ruelle entropy inequality.

(2). Note that SRB is equivalent to . So if and only if , which also equivalent to (by Ledrappier-Young) and hence .

(3). Pick . For each , there exists with . So , where . Clearly and hence as . In particular .