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Nonanalyticities in the generalized dimensions of fractal sets of physical interest are interpreted as phase transitions. We apply the thermodynamical formalism to the fractal set formed by the irrational winding parameter values of critical circle maps and introduce and investigate in detail several distinct fractal measures on this set. The thermodynamic functions associated with different measures are distinct: We discover that, in all cases that we study, they exhibit phase transitions. The numerical estimates of the Hausdorff dimension from various versions of the thermodynamical formalism and a variety of circle maps yield D H= 0.8701±0.0003 and are consistent with the conjectured universality of D H.