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We show that, generically, the unique invariant measure of a sufficientlyregular piecewise smooth circle homeomorphism with irrational rotation numberand zero mean nonlinearity (e.g., piecewise linear) has zero Hausdorffdimension.  To encode this generic condition, we consider piecewise smooth homeomorphismsas generalized interval exchange transformations (GIETs) of the interval andrely on the notion of combinatorial rotation number for GIETs, which can beseen as an extension of the classical notion of rotation number for circlehomeomorphisms to the GIET setting.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms