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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 |
References:
[1] |
W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps,, Nonlinearity, 25 (2012), 107.
doi: 10.1088/0951-7715/25/1/107. |
[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V. Baladi, Unpublished,, 1996., ().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. |
[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. |
[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. |
[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. |
[7] |
M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012).
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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). 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. |
[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. |
[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. |
[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. |
[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. |
[25] |
F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119.
doi: 10.1007/BF01215004. |
[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. |
[27] |
T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators,, J. Analyse Math., 6 (1958), 261.
doi: 10.1007/BF02790238. |
[28] |
T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).
|
[29] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.
|
[37] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.
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.
doi: 10.1006/jcph.1999.6231. |
[39] |
R. S. Strichartz, Multipliers on fractional Sobolev spaces,, J. Math. Mech., 16 (1967), 1031.
|
[40] |
J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012).
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.
doi: 10.1017/S0143385700001012. |
[42] |
S. M. Ulam, A Collection of Mathematical Problems,, Interscience Tracts in Pure and Applied Mathematics, (1960).
|
show all references
References:
[1] |
W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps,, Nonlinearity, 25 (2012), 107.
doi: 10.1088/0951-7715/25/1/107. |
[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. |
[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. |
[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. |
[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. |
[7] |
M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012).
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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). 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. |
[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. |
[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. |
[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. |
[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. |
[25] |
F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119.
doi: 10.1007/BF01215004. |
[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. |
[27] |
T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators,, J. Analyse Math., 6 (1958), 261.
doi: 10.1007/BF02790238. |
[28] |
T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).
|
[29] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
M. Rychlik, Bounded variation and invariant measures,, Studia Math., 76 (1983), 69.
|
[37] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps,, Israel J. Math., 116 (2000), 223.
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.
doi: 10.1006/jcph.1999.6231. |
[39] |
R. S. Strichartz, Multipliers on fractional Sobolev spaces,, J. Math. Mech., 16 (1967), 1031.
|
[40] |
J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012).
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.
doi: 10.1017/S0143385700001012. |
[42] |
S. M. Ulam, A Collection of Mathematical Problems,, Interscience Tracts in Pure and Applied Mathematics, (1960).
|
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