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We calculate the value of the average of the singularities of a Gibbs measure μ invariant with respect to an expansive C 2 diffeomorphism of a one-compact manifold. This is the value related to dimension that one computes numerically. We then define and study a function, known as the correlation dimension, which is related to a free energy function, and we generalize the results in higher dimension with an axiom A transformation acting on a two-compact manifold.

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