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Irrational rotation numbers

For a general class of one-parameter families of 'flat spot' circle maps (such as non-decreasing truncations of non-invertible circle maps), the author proves the following facts. The set of parameter values where the rotation number is irrational has Hausdorff dimension zero. Each recurrent set with irrational rotation number has Hausdorff dimension zero. Moreover, the closure of their union has Lebesgue measure zero. The author's results are less general than the ones obtained by Swiatek (1988) however, they are stronger and the proofs are much simpler.