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.
Published inDiscrete and Continuous Dynamical Systems
Pages529 - 540
CitationBAHSOUN, W., BOSE, C. and QUAS, A., 2008. Deterministic representation for position-dependent random maps. Discrete and Continuous Dynamical Systems. Series A, 22 (3), pp.529-540.
Publisher© American Institute of Mathematical Sciences
VersionAM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesThis is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems, Series A following peer review. The definitive publisher-authenticated version, BAHSOUN, W., BOSE, C. and QUAS, A., 2008. Deterministic representation for position-dependent random maps. Discrete and Continuous Dynamical Systems. Series A, 22 (3), pp.529-540, is available online at: https://doi.org/10.3934/dcds.2008.22.529.