Minimum 'energy' approximations of invariant measures for nonsingular transformations

Previous Article
Time-periodic solutions of the Boltzmann equation
DCDS Home
This Issue
Next Article

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

Minimum 'energy' approximations of invariant measures for nonsingular transformations

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

Abstract
Related Papers

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

[1]	Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235
[2]	Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
[3]	Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007
[4]	Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015
[5]	Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457
[6]	Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
[7]	Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 187-207. doi: 10.3934/dcds.2011.30.187
[8]	Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016
[9]	Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019032
[10]	Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[11]	Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[12]	Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[13]	Abraão D. C. Nascimento, Leandro C. Rêgo, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036
[14]	Richard Miles, Michael Björklund. Entropy range problems and actions of locally normal groups. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 981-989. doi: 10.3934/dcds.2009.25.981
[15]	Donald Ornstein, Benjamin Weiss. Entropy is the only finitely observable invariant. Journal of Modern Dynamics, 2007, 1 (1) : 93-105. doi: 10.3934/jmd.2007.1.93
[16]	Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1
[17]	Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[18]	Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
[19]	Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.
[20]	Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic & Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences