In many situations, both deterministic and probabilistic, one can develop further the study of the multifractal structure of a dynamical system, particularly when there exist strange attractors. Multifractal refers to a notion of size emphasizing the variations of the weigth of the measure. In such schemes, one has to compute a free energy function associated to some sequence of partitions. We relate the free energy function, associated to a sequence of uniform partitions of exponentially decreasing diameters, and the correlation dimension which refers to a quantity that is the most accessible in numerical computations. Finally we discuss of two assumptions for the existence of free energy functions.