Ambiguity in the determination of the free energy associated with the critical circle map

We consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.