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self homeomorphism of circle

Source: miklos schweitzer 2006 q9

September 3, 2021
topologyhomeomorphismabsolute continuity

Problem Statement

Does the circle T = R / Z have a self-homeomorphism ϕ\phi that is singular (that is, its derivative is almost everywhere 0), but the mapping f:T→Tf:T \to T , f(x)=ϕ−1(2ϕ(x))f(x) = \phi^{-1} (2\phi(x)) is absolutely continuous?
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