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A FAMILY OF RANDOM MAPS WHICH POSSES INFINITE ABSOLUTELY CONTINUOUS INVARIANT MEASURES

ABSTRACT: We consider a family of random maps where each of the component maps is from a family of piecewise, linear and Markov maps on a class of infinite partitions of the state space. We investigate the existence of infinite absolutely continuous invariant measures of the random maps. In a more general setting, our study establishes a positive answer to the question in discrete time dynamical system: can two chaotic systems give rise to order, namely can they be combined into another dynamical system which does not behave chaotically? This question is analogous to Parrondo’s paradox [5] which states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game = a winning game.