Geodesic
Contraction of Orbits in Random Dynamical Systems on the Circle
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 36-54

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The paper deals with a theoretical justification of the effect, observed in computer experiments, of convergence of orbits (without tending to any particular point) in random dynamical systems on the circle. The corresponding theorem is proved under certain assumptions satisfied, in particular, in some $C^1$-open domain in the space of random dynamical systems. It follows from this theorem that the corresponding skew product has two invariant measurable sections, naturally called an attractor and a repeller. Moreover, it turns out that convergence of orbits and the uniqueness of a stationary measure, phenomena that are mutually exclusive in the case of a single map, typically coexist in random dynamical systems.
Keywords: dynamics on the circle, random dynamical system, skew product, attractor. 
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     author = {V. A. Kleptsyn and M. B. Nalsky},
     title = {Contraction of {Orbits} in {Random} {Dynamical} {Systems} on the {Circle}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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V. A. Kleptsyn; M. B. Nalsky. Contraction of Orbits in Random Dynamical Systems on the Circle. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 36-54. http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a4/