Example of homeomorphism of $S^1$ which behaves badly relatively to the Lebesgue measure Could anyone give an example of a homeomorphism of $S^1$ which sends an open set of full Lebesgue measure on an open set which has not full measure ?
 A: Such maps abound and, moreover, they appear very naturally. Recall that a map of two measure spaces is said to have Luzin's Property N if it sends each set of zero measure to a set of zero measure. In your setting (of homeomorphisms $f: S^1\to S^1$) Luzin's Property N is equivalent to absolute continuity of $f^{-1}$. Examples of homeomorphisms $S^1\to S^1$ which are not absolutely continuous appear naturally in the Teichmuller Theory: Suppose that $S_1, S_2$ are two compact hyperbolic surfaces of the same genus, $S_i=H^2/\Gamma_i$, where $\Gamma_i< PSL(2,R)$ is a Fuchsian subgroup. Then there exists a quasi-symmetric homeomorphisms $f: S^1\to S^1$ equivariant with respect to the isomorphism $\Gamma_1\to \Gamma_2$ (induced by a homeomorphism $S_1\to S_2$). 
Theorem. $f$ is an element of $PLS(2,R)$ if and only if it is absolutely continuous. 
See this paper by Agard: "Remarks on the boundary mapping for a Fuchsian group". In fact, Agard proves much more: If $f$ has nonzero derivative at one point, then $f\in PSL(2,R)$. 
