An elementary way to rigorously estimate convergence to equilibrium and escape rates

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January 2015, 2(1): 51-64. doi: 10.3934/jcd.2015.2.51

An elementary way to rigorously estimate convergence to equilibrium and escape rates

Stefano Galatolo ^1, , Isaia Nisoli ^2, and Benoît Saussol ^3,

1.	Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano
2.	Instituto de Matemática, UFRJ Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C Cidade Universitária, Ilha do Fundão, Caixa Postal 68530 21941-909 Rio de Janeiro, RJ, Brazil
3.	Laboratoire de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 av. Victor Le Gorgeu, CS 93837, 29238 BREST Cedex 3

Received April 2014 Revised January 2015 Published August 2015

Abstract
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We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.

Keywords: interval arithmetics, Ulam method., Rigorous computation, decay of correlation, convergence to equilibrium.

Mathematics Subject Classification: Primary: 37M25, 65G3.

Citation: Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51

References:

[1]	V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Mathematiche Zeitcrift, 276 (2014), 1001-1048. doi: 10.1007/s00209-013-1231-0.
[2]	W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proceedings in Mathematics & Statistics, 70. Springer, New York, 2014. doi: 10.1007/978-1-4939-0419-8.
[3]	W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542. doi: 10.1088/0951-7715/19/11/002.
[4]	W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$, Nonlinear Analysis, 74 (2011), 4481-4495. doi: 10.1016/j.na.2011.04.012.
[5]	V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators, Nonlinearity Nonlinearity, 12 (1999), 525-538. doi: 10.1088/0951-7715/12/3/006.
[6]	L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427.
[7]	C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes, arXiv:1204.2329v2
[8]	M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.
[9]	M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.
[10]	G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001.
[11]	G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257. doi: 10.1007/s002200050198.
[12]	S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures, SIAM J. Appl Dyn Sys., 13 (2014), 958-985. doi: 10.1137/130911044.
[13]	S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805. doi: 10.4310/MRL.2007.v14.n5.a8.
[14]	S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps, arXiv:1402.5918 (to appear on Erg. Th. Dyn. Sys.).
[15]	B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749. doi: 10.1017/S014338570000907X.
[16]	G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152.
[17]	O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816. doi: 10.1137/09077864X.
[18]	O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps, Exp. Math., 7 (1998), 317-324. doi: 10.1080/10586458.1998.10504377.
[19]	A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.
[20]	C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study, Nonlinearity, 14 (2001), 463-490. doi: 10.1088/0951-7715/14/3/303.
[21]	C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004.

show all references

References:

[1]	V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Mathematiche Zeitcrift, 276 (2014), 1001-1048. doi: 10.1007/s00209-013-1231-0.
[2]	W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proceedings in Mathematics & Statistics, 70. Springer, New York, 2014. doi: 10.1007/978-1-4939-0419-8.
[3]	W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542. doi: 10.1088/0951-7715/19/11/002.
[4]	W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$, Nonlinear Analysis, 74 (2011), 4481-4495. doi: 10.1016/j.na.2011.04.012.
[5]	V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators, Nonlinearity Nonlinearity, 12 (1999), 525-538. doi: 10.1088/0951-7715/12/3/006.
[6]	L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427.
[7]	C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes, arXiv:1204.2329v2
[8]	M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.
[9]	M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.
[10]	G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001.
[11]	G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257. doi: 10.1007/s002200050198.
[12]	S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures, SIAM J. Appl Dyn Sys., 13 (2014), 958-985. doi: 10.1137/130911044.
[13]	S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805. doi: 10.4310/MRL.2007.v14.n5.a8.
[14]	S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps, arXiv:1402.5918 (to appear on Erg. Th. Dyn. Sys.).
[15]	B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749. doi: 10.1017/S014338570000907X.
[16]	G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152.
[17]	O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816. doi: 10.1137/09077864X.
[18]	O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps, Exp. Math., 7 (1998), 317-324. doi: 10.1080/10586458.1998.10504377.
[19]	A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.
[20]	C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study, Nonlinearity, 14 (2001), 463-490. doi: 10.1088/0951-7715/14/3/303.
[21]	C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004.

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