On Absolutely Continuous Invariant Measures of Noncontracting Transformations of a Circle

A result reported earlier by the authors is described in detail. An existence condition is obtained for an absolutely continuous invariant measure for (locally) noncontracting mappings of an interval and a circle. This condition does not require the monotonicity of the derivative of the mappings in neighborhoods of their nonhyperbolic fixed points. It is proved that a noncontracting $\mathrm C^2$ mapping $f$ of a circle into itself which is nonflat at the points where $f'=1$ admits an absolutely continuous infinite invariant measure. It is shown that the constraint on the class of smoothness cannot be weakened.