Ambiguity in the Determination of the Free Energy for a Model of the   Circle Map
  We consider a simple model to describe the widths of the mode locked intervals for the critical circle map. 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 is found that the result of the calculation depends on the method of partition. An implication of this is that the generalized dimensions $D_q$ are different for each partition except when $q=0$, i.e. only the Hausdorff dimension is the same in each case. 
Brian G. Kenny and Tony W. Dixon