Rotation number, invariant measures and ratio set of PL homeomorphisms of the circle.
The paper deals with piecewise affine circle homeomorphisms without periodic points. This class of maps was studied by M. Herman who found examples of transformations of this class with singular invariant measures for any rotation number. On the other hand, it is not difficult to find examples in which points of discontinuity of the derivative meet other such points and the jumps of the derivative cancel out. The present paper shows that, at least under some circumstances, this is the only way to get an absolutely continuous invariant measure, whose density then turns out to be piecewise analytic. The key condition is that that in order for the conjugacy to be piecewise \(C^1\), the number of points of non-differentiability must remain uniformly bounded for all iterates of the map. It would be interesting to know whether any of these results remain true for mappings which are piecewise smooth, but not affine, and for which the product of all jumps of the derivative is 1.
scientific article; zbMATH DE number 2171514
scientific article