|A FAMILY OF RANDOM MAPS WHICH POSSES INFINITE ABSOLUTELY CONTINUOUS INVARIANT MEASURES
|Year of Publication
|Dynamic Systems and Applications
|AMS Subject Classification
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  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.
 J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, American Mathematical Society, 1997.