Geodesic
On Measure Invariance for a 2-valued Transformation
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 183-191

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We consider a family of 2-valued transformations of special form on the interval $[0,1]$ with measure $\mu=\int p(x)\,d\lambda$ which is absolutely continuous with respect to the Lebesgue measure. We endow $S$ with a set of weight functions $\alpha=\{\alpha_1(x),\alpha_2(x)\}$ and find a criterion of measure invariance under the transformation. This criterion relates the three parameters $a$, $p$, $\alpha$ to each other.
Keywords: multivalued dynamical system, invariant measure. 
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     title = {On {Measure} {Invariance} for a~2-valued {Transformation}},
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P. I. Troshin. On Measure Invariance for a 2-valued Transformation. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 183-191. http://geodesic.mathdoc.fr/item/UZKU_2009_151_4_a15/