Let be the space of symbols, be an matrix with , be the set of sequences that is -admissible. Consider the dynamical system We assume this system is mixing.

Let be a Holder potential, which induces a transfer operator on the space of continuous functions: .

Let be the spectral radius of . Then is also an eigenvalue of , which is called the principle eigenvalue. Moreover, there exists a positive eigenfunction such that . Replacing by , we will assume .

Consider the conjugate action on the space of functional (or sign measures). There is a positive eigenmeasure such that .

We normalize the pair such that . Then the measure is a -invariant probability measure. It is called the equilibrium state of .

Two continuous functions is called cohomologous if there exists a continuous function such that

.

Let be cohomologous. Then the two operators and are different, but .

Their eigenfunctions and eigenmeasures are different, but the associated equilibrium states are the same.

To find a natural representative in the class of functions that are cohomologous to , we set . Then we have

1). . So is the eigenfunction of .

2). .

So is the eigenmeasure of .

From this point of view, we might pick as the representative of .