We prove that a critical circle homeomorphism with infinite number of break points without periodic orbits is conjugated to the linear rotation by a quasisymmetric map if and only if its rotation number is of bounded type. And we also prove that any two adjacent atoms of dynamical partition of a unit circle are comparable.
"On Critical Circle Homeomorphisms with Infinite Number of Break Points." Abstr. Appl. Anal. 2014 1 - 6, 2014. https://doi.org/10.1155/2014/378742