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Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point



We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$ , the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$ -smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$ , with rotation number $\unicode[STIX]{x1D70C}$ , is zero. This result cannot be extended to all irrational rotation numbers.



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