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September  2008, 22(3): 529-540. doi: 10.3934/dcds.2008.22.529

Deterministic representation for position dependent random maps

Wael Bahsoun 1, , Christopher Bose 2, and  Anthony Quas 3,

1. 

Economics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
2. 

Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC, Canada V8W 3P4
3. 

Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4, Canada

Received  July 2007 Revised  February 2008 Published  August 2008

We give a deterministic representation for position dependent random maps and describe the structure of its set of invariant measures. Our construction generalizes the skew product representation of random maps with constant probabilities. In particular, we establish one-to-one correspondence between eigenfunctions corresponding to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator of the random map and for those of the skew. An immediate consequence is one-to-one correspondence between absolutely continuous invariant measures (acims) for the position dependent random map and acims for its deterministic representation.
Keywords: Random map, skew product, absolutely continuous invariant measure..
Mathematics Subject Classification: Primary 37A05, 37E0.
Citation: Wael Bahsoun, Christopher Bose, Anthony Quas. Deterministic representation for position dependent random maps. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 529-540. doi: 10.3934/dcds.2008.22.529
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