Geodesic
On Absolutely Continuous Invariant Measures of Noncontracting Transformations of a Circle
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 23-34

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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. 
@article{TM_2004_244_a2,
     author = {Sh. I. Akhalaya and A. M. Stepin},
     title = {On {Absolutely} {Continuous} {Invariant} {Measures} of {Noncontracting} {Transformations} of a {Circle}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {23--34},
     publisher = {mathdoc},
     volume = {244},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_244_a2/}
}
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Sh. I. Akhalaya; A. M. Stepin. On Absolutely Continuous Invariant Measures of Noncontracting Transformations of a Circle. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 23-34. http://geodesic.mathdoc.fr/item/TM_2004_244_a2/