Physica D 210 (2005) 284–294 Randomly chosen chaotic maps can give rise to nearly ordered behavior Abraham Boyarsky, Pawel G´ ora ∗ , Md. Shafiqul Islam 1 Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Que., Canada H4B 1R6 Received 21 January 2005; received in revised form 21 July 2005; accepted 25 July 2005 Available online 15 August 2005 Communicated by R. Roy Abstract Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226–5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72]) 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. Inspired by this paradox, a recent study [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] asked an analogous 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? Numerical evidence is provided in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] that two chaotic quadratic maps, when composed with each other, create a new dynamical system which has a stable period orbit. The question of what happens in the case of random composition of maps is posed in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] but left unanswered. In this note we present an example of a dynamical system where, at each iteration, a map is chosen in a probabilistic manner from a collection of chaotic maps. The resulting random map is proved to have an infinite absolutely continuous invariant measure (acim) with spikes at two points. From this we show that the dynamics behaves in a nearly ordered manner. When the foregoing maps are applied one after the other, deterministically as in [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72], the resulting composed map has a periodic orbit which is stable. © 2005 Elsevier B.V. All rights reserved. Keywords: Infinite measure; Absolutely continuous invariant measure; Parrando’s paradox; Random maps; Invariant measures; Ergodicity; Absolutely continuous measures; Frobenius–Perron operator ∗ Corresponding author. E-mail addresses: boyar@alcor.concordia.ca (A. Boyarsky), pgora@vax2.concordia.ca (P. G´ ora), islam@cs.uleth.ca (M.S. Islam). 1 Present address: Department of Mathematics and Computer Science, University Hall, 4401 University Drive, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4. 0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2005.07.015