American Institute of Mathematical Sciences

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July  2006, 14(3): 597-615. doi: 10.3934/dcds.2006.14.597

Minimum 'energy' approximations of invariant measures for nonsingular transformations

Christopher Bose 1, and  Rua Murray 2,

1. 

Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC, Canada V8W 3P4
2. 

Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton

Received  October 2004 Revised  May 2005 Published  December 2005

We study variational methods for rigorous approximation of invariant densities for a nonsingular map $T$ on a Borel measure space. The general method takes the form of a convergent sequence of optimization problems on $L^p$, $1 \leq p < \infty$ with a convex objective and finite moment constraints. Provided $T$ admits an invariant density in the appropriate $L^p$ space, weak convergence of the sequence of optimal solutions is observed; norm convergence can be obtained when the objective is a Kadec functional. No regularity or expansiveness assumptions on $T$ need to be made, and the method applies to maps on multidimensional domains. Objectives leading to norm convergence include Entropy, 'Energy' and 'Positively Constrained Energy'.
   Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.
Keywords: Frobenius-Perron operator, entropy, normal convex integrand, duality., Kadec, moment problem, Invariant measure.
Mathematics Subject Classification: Primary: 28D05; Secondary: 37M25, 41A46, 49K2.
Citation: Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
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