# American Institute of Mathematical Sciences

June  2014, 1(2): 249-278. doi: 10.3934/jcd.2014.1.249

## Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools

 1 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia 3 Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4

Received  October 2013 Revised  October 2014 Published  December 2014

The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the efficacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.
Citation: Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249
##### References:
 [1] W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps, Nonlinearity, 25 (2012), 107-124. doi: 10.1088/0951-7715/25/1/107. [2] V. Baladi, Unpublished, 1996. [3] V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. [4] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, in Algebraic and topological dynamics, vol. 385 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2005), 123-135. doi: 10.1090/conm/385/07194. [5] V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1453-1481. doi: 10.1016/j.anihpc.2009.01.001. [6] M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973. doi: 10.1088/0951-7715/15/6/309. [7] M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510. doi: 10.1063/1.4772195. [8] J. Buzzi, No or infinitely many a.c.i.p. for piecewise expanding $C^r$ maps in higher dimensions, Comm. Math. Phys., 222 (2001), 495-501. doi: 10.1007/s002200100509. [9] W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory Dynam. Systems, 22 (2002), 1061-1078. doi: 10.1017/S0143385702000627. [10] M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188. doi: 10.1088/0951-7715/13/4/310. [11] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [12] M. F. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814. doi: 10.1090/S0002-9947-08-04464-4. [13] D. Dolgopyat and P. Wright, The diffusion coefficient for piecewise expanding maps of the interval with metastable states, Stoch. Dyn., 12 (2012), 1150005, 13pp. doi: 10.1142/S0219493712003547. [14] G. Froyland, R. Murray and O. Stancevic, Spectral degeneracy and escape dynamics for intermittent maps with a hole, Nonlinearity, 24 (2011), 2435-2463. doi: 10.1088/0951-7715/24/9/003. [15] G. Froyland, Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps, Phys. D, 237 (2008), 840-853. doi: 10.1016/j.physd.2007.11.004. [16] G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860. doi: 10.3934/dcds.2013.33.3835. [17] G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002. [18] G. Froyland, K. Padberg, M. England and A.-M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007), 224503. [19] G. Froyland and O. Stancevic, Escape rates and Perron-Frobenius operators: Open and closed dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 457-472. doi: 10.3934/dcdsb.2010.14.457. [20] C. González-Tokman, B. Hunt and P. Wright, Approximating invariant densities of metastable systems, Ergodic Theory and Dynamical Systems, 31 (2011), 1345-1361. doi: 10.1017/S0143385710000337. [21] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272, URL http://journals.cambridge.org/article_S0143385712001897. doi: 10.1017/etds.2012.189. [22] P. Góra, A. Boyarsky and H. Proppe, On the number of invariant measures for higher-dimensional chaotic transformations, J. Statist. Phys., 62 (1991), 709-728. doi: 10.1007/BF01017979. [23] G. Gripenberg, Fourier Analysis, 2009, Lecture Notes. [24] H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348. [25] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004. [26] O. Junge, J. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps, in Decision and Control, 2004. CDC. 43rd IEEE Conference on, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379. [27] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math., 6 (1958), 261-322. doi: 10.1007/BF02790238. [28] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. [29] G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103941781. doi: 10.1007/BF01240219. [30] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152, URL http://www.numdam.org/item?id=ASNSP_1999_4_28_1_141_0. [31] G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, J. Stat. Phys., 135 (2009), 519-534. doi: 10.1007/s10955-009-9747-8. [32] G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730. doi: 10.1088/0951-7715/17/5/009. [33] Z. Levnajić and I. Mezić, Ergodic theory and visualization. i. mesochronic plots for visualization of ergodic partition and invariant sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 033114, 19pp. doi: 10.1063/1.3458896. [34] G. Mathew, I. Mezić and L. Petzold, A multiscale measure for mixing, Phys. D, 211 (2005), 23-46. doi: 10.1016/j.physd.2005.07.017. [35] I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D: Nonlinear Phenomena, 197 (2004), 101-133, URL http://www.sciencedirect.com/science/article/pii/S0167278904002507. doi: 10.1016/j.physd.2004.06.015. [36] M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. [37] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219. [38] C. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on hybrid Monte Carlo, J. Comput. Phys., 151 (1999), 146-168, Computational molecular biophysics. doi: 10.1006/jcph.1999.6231. [39] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [40] J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity, 25 (2012), R1-R44. doi: 10.1088/0951-7715/25/2/R1. [41] M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory Dynam. Systems, 20 (2000), 1851-1857. doi: 10.1017/S0143385700001012. [42] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960.

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##### References:
 [1] W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps, Nonlinearity, 25 (2012), 107-124. doi: 10.1088/0951-7715/25/1/107. [2] V. Baladi, Unpublished, 1996. [3] V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. [4] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, in Algebraic and topological dynamics, vol. 385 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2005), 123-135. doi: 10.1090/conm/385/07194. [5] V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1453-1481. doi: 10.1016/j.anihpc.2009.01.001. [6] M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973. doi: 10.1088/0951-7715/15/6/309. [7] M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510. doi: 10.1063/1.4772195. [8] J. Buzzi, No or infinitely many a.c.i.p. for piecewise expanding $C^r$ maps in higher dimensions, Comm. Math. Phys., 222 (2001), 495-501. doi: 10.1007/s002200100509. [9] W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory Dynam. Systems, 22 (2002), 1061-1078. doi: 10.1017/S0143385702000627. [10] M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188. doi: 10.1088/0951-7715/13/4/310. [11] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [12] M. F. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814. doi: 10.1090/S0002-9947-08-04464-4. [13] D. Dolgopyat and P. Wright, The diffusion coefficient for piecewise expanding maps of the interval with metastable states, Stoch. Dyn., 12 (2012), 1150005, 13pp. doi: 10.1142/S0219493712003547. [14] G. Froyland, R. Murray and O. Stancevic, Spectral degeneracy and escape dynamics for intermittent maps with a hole, Nonlinearity, 24 (2011), 2435-2463. doi: 10.1088/0951-7715/24/9/003. [15] G. Froyland, Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps, Phys. D, 237 (2008), 840-853. doi: 10.1016/j.physd.2007.11.004. [16] G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860. doi: 10.3934/dcds.2013.33.3835. [17] G. Froyland and K. Padberg, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), 1507-1523. doi: 10.1016/j.physd.2009.03.002. [18] G. Froyland, K. Padberg, M. England and A.-M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007), 224503. [19] G. Froyland and O. Stancevic, Escape rates and Perron-Frobenius operators: Open and closed dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 457-472. doi: 10.3934/dcdsb.2010.14.457. [20] C. González-Tokman, B. Hunt and P. Wright, Approximating invariant densities of metastable systems, Ergodic Theory and Dynamical Systems, 31 (2011), 1345-1361. doi: 10.1017/S0143385710000337. [21] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory and Dynamical Systems, 34 (2014), 1230-1272, URL http://journals.cambridge.org/article_S0143385712001897. doi: 10.1017/etds.2012.189. [22] P. Góra, A. Boyarsky and H. Proppe, On the number of invariant measures for higher-dimensional chaotic transformations, J. Statist. Phys., 62 (1991), 709-728. doi: 10.1007/BF01017979. [23] G. Gripenberg, Fourier Analysis, 2009, Lecture Notes. [24] H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348. [25] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004. [26] O. Junge, J. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps, in Decision and Control, 2004. CDC. 43rd IEEE Conference on, 2 (2004), 2225-2230. doi: 10.1109/CDC.2004.1430379. [27] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math., 6 (1958), 261-322. doi: 10.1007/BF02790238. [28] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. [29] G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103941781. doi: 10.1007/BF01240219. [30] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152, URL http://www.numdam.org/item?id=ASNSP_1999_4_28_1_141_0. [31] G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae, J. Stat. Phys., 135 (2009), 519-534. doi: 10.1007/s10955-009-9747-8. [32] G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730. doi: 10.1088/0951-7715/17/5/009. [33] Z. Levnajić and I. Mezić, Ergodic theory and visualization. i. mesochronic plots for visualization of ergodic partition and invariant sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20 (2010), 033114, 19pp. doi: 10.1063/1.3458896. [34] G. Mathew, I. Mezić and L. Petzold, A multiscale measure for mixing, Phys. D, 211 (2005), 23-46. doi: 10.1016/j.physd.2005.07.017. [35] I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D: Nonlinear Phenomena, 197 (2004), 101-133, URL http://www.sciencedirect.com/science/article/pii/S0167278904002507. doi: 10.1016/j.physd.2004.06.015. [36] M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. [37] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248. doi: 10.1007/BF02773219. [38] C. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on hybrid Monte Carlo, J. Comput. Phys., 151 (1999), 146-168, Computational molecular biophysics. doi: 10.1006/jcph.1999.6231. [39] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [40] J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity, 25 (2012), R1-R44. doi: 10.1088/0951-7715/25/2/R1. [41] M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory Dynam. Systems, 20 (2000), 1851-1857. doi: 10.1017/S0143385700001012. [42] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960.
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