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MFAT 15 (2009), no. 1, 15-30
We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous.