# 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:
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Wright, Approximating invariant densities of metastable systems,, Ergodic Theory and Dynamical Systems, 31 (2011), 1345.  doi: 10.1017/S0143385710000337.  Google Scholar [21] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem,, Ergodic Theory and Dynamical Systems, 34 (2014), 1230.  doi: 10.1017/etds.2012.189.  Google Scholar [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.  doi: 10.1007/BF01017979.  Google Scholar [23] G. Gripenberg, Fourier Analysis, 2009,, Lecture Notes., ().   Google Scholar [24] H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens,, Proc. Amer. Math. Soc., 118 (1993), 627.  doi: 10.2307/2160348.  Google Scholar [25] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119.  doi: 10.1007/BF01215004.  Google Scholar [26] O. Junge, J. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps,, in Decision and Control, 2 (2004), 2225.  doi: 10.1109/CDC.2004.1430379.  Google Scholar [27] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators,, J. Analyse Math., 6 (1958), 261.  doi: 10.1007/BF02790238.  Google Scholar [28] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar [29] G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.  doi: 10.1007/BF01240219.  Google Scholar [30] G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar [31] G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae,, J. Stat. Phys., 135 (2009), 519.  doi: 10.1007/s10955-009-9747-8.  Google Scholar [32] G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps,, Nonlinearity, 17 (2004), 1723.  doi: 10.1088/0951-7715/17/5/009.  Google Scholar [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).  doi: 10.1063/1.3458896.  Google Scholar [34] G. Mathew, I. Mezić and L. Petzold, A multiscale measure for mixing,, Phys. D, 211 (2005), 23.  doi: 10.1016/j.physd.2005.07.017.  Google Scholar [35] I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.  doi: 10.1016/j.physd.2004.06.015.  Google Scholar [36] M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.   Google Scholar [37] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.  doi: 10.1007/BF02773219.  Google Scholar [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.  doi: 10.1006/jcph.1999.6231.  Google Scholar [39] R. S. Strichartz, Multipliers on fractional Sobolev spaces,, J. Math. Mech., 16 (1967), 1031.   Google Scholar [40] J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012).  doi: 10.1088/0951-7715/25/2/R1.  Google Scholar [41] M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties,, Ergodic Theory Dynam. Systems, 20 (2000), 1851.  doi: 10.1017/S0143385700001012.  Google Scholar [42] S. M. Ulam, A Collection of Mathematical Problems,, Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar

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##### References:
 [1] W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps,, Nonlinearity, 25 (2012), 107.  doi: 10.1088/0951-7715/25/1/107.  Google Scholar [2] V. Baladi, Unpublished,, 1996., ().   Google Scholar [3] V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics,, World Scientific Publishing Co. Inc., (2000).  doi: 10.1142/9789812813633.  Google Scholar [4] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations,, in Algebraic and topological dynamics, (2005), 123.  doi: 10.1090/conm/385/07194.  Google Scholar [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.  doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar [6] M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps,, Nonlinearity, 15 (2002), 1905.  doi: 10.1088/0951-7715/15/6/309.  Google Scholar [7] M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012).  doi: 10.1063/1.4772195.  Google Scholar [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.  doi: 10.1007/s002200100509.  Google Scholar [9] W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps,, Ergodic Theory Dynam. Systems, 22 (2002), 1061.  doi: 10.1017/S0143385702000627.  Google Scholar [10] M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator,, Nonlinearity, 13 (2000), 1171.  doi: 10.1088/0951-7715/13/4/310.  Google Scholar [11] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar [12] M. F. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777.  doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar [13] D. Dolgopyat and P. Wright, The diffusion coefficient for piecewise expanding maps of the interval with metastable states,, Stoch. Dyn., 12 (2012).  doi: 10.1142/S0219493712003547.  Google Scholar [14] G. Froyland, R. Murray and O. Stancevic, Spectral degeneracy and escape dynamics for intermittent maps with a hole,, Nonlinearity, 24 (2011), 2435.  doi: 10.1088/0951-7715/24/9/003.  Google Scholar [15] G. Froyland, Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps,, Phys. D, 237 (2008), 840.  doi: 10.1016/j.physd.2007.11.004.  Google Scholar [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.  doi: 10.3934/dcds.2013.33.3835.  Google Scholar [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.  doi: 10.1016/j.physd.2009.03.002.  Google Scholar [18] G. Froyland, K. Padberg, M. England and A.-M. Treguier, Detection of coherent oceanic structures via transfer operators,, Phys. Rev. Lett., 98 (2007).   Google Scholar [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.  doi: 10.3934/dcdsb.2010.14.457.  Google Scholar [20] C. González-Tokman, B. Hunt and P. Wright, Approximating invariant densities of metastable systems,, Ergodic Theory and Dynamical Systems, 31 (2011), 1345.  doi: 10.1017/S0143385710000337.  Google Scholar [21] C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem,, Ergodic Theory and Dynamical Systems, 34 (2014), 1230.  doi: 10.1017/etds.2012.189.  Google Scholar [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.  doi: 10.1007/BF01017979.  Google Scholar [23] G. Gripenberg, Fourier Analysis, 2009,, Lecture Notes., ().   Google Scholar [24] H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens,, Proc. Amer. Math. Soc., 118 (1993), 627.  doi: 10.2307/2160348.  Google Scholar [25] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119.  doi: 10.1007/BF01215004.  Google Scholar [26] O. Junge, J. Marsden and I. Mezic, Uncertainty in the dynamics of conservative maps,, in Decision and Control, 2 (2004), 2225.  doi: 10.1109/CDC.2004.1430379.  Google Scholar [27] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators,, J. Analyse Math., 6 (1958), 261.  doi: 10.1007/BF02790238.  Google Scholar [28] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).   Google Scholar [29] G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.  doi: 10.1007/BF01240219.  Google Scholar [30] G. Keller and C. Liverani, Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141.   Google Scholar [31] G. Keller and C. Liverani, Rare events, escape rates and quasistationarity: Some exact formulae,, J. Stat. Phys., 135 (2009), 519.  doi: 10.1007/s10955-009-9747-8.  Google Scholar [32] G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps,, Nonlinearity, 17 (2004), 1723.  doi: 10.1088/0951-7715/17/5/009.  Google Scholar [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).  doi: 10.1063/1.3458896.  Google Scholar [34] G. Mathew, I. Mezić and L. Petzold, A multiscale measure for mixing,, Phys. D, 211 (2005), 23.  doi: 10.1016/j.physd.2005.07.017.  Google Scholar [35] I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.  doi: 10.1016/j.physd.2004.06.015.  Google Scholar [36] M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.   Google Scholar [37] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.  doi: 10.1007/BF02773219.  Google Scholar [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.  doi: 10.1006/jcph.1999.6231.  Google Scholar [39] R. S. Strichartz, Multipliers on fractional Sobolev spaces,, J. Math. Mech., 16 (1967), 1031.   Google Scholar [40] J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012).  doi: 10.1088/0951-7715/25/2/R1.  Google Scholar [41] M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties,, Ergodic Theory Dynam. Systems, 20 (2000), 1851.  doi: 10.1017/S0143385700001012.  Google Scholar [42] S. M. Ulam, A Collection of Mathematical Problems,, Interscience Tracts in Pure and Applied Mathematics, (1960).   Google Scholar
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