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Stability of the spectrum for transfer operators. (English) Zbl 0956.37003

One of the long standing and broadly discussed problems in chaotic dynamics is the problem of stochastic stability, i.e. the stability of transfer (Perron-Frobenius) operators of chaotic dynamical systems with respect to various small random (or deterministic) perturbations. Usually this problem is considered in terms of the stability of the density of the so called Sinai-Bowen-Ruelle measure. This density (if it is well defined) is the leading eigenfunction of the corresponding transfer operator considered in a certain Banach space (for example, in the space of functions of bounded variation). More generally one can think about this problem as the problem of stability (instability) of the complete spectrum of this operator.
In this context the analysis of the stochastic stability problem was started by V. Baladi and L. S. Young [Commun. Math. Phys. 156, 355-385 (1993; Zbl 0809.60101)] for the case of convolution type random perturbations of smooth expanding one-dimensional maps. Later for general random and Ulam-type perturbations of piecewise expanding maps this problem was studied by M. Blank and G. Keller [Nonlinearity 11, 1351-1364 (1998; Zbl 0921.58051)]. It is worth noticing that while one can find sufficient conditions for the stochastic stability of isolated eigenvalues for certain classes of transfer operators, no conditions for stability of the essential spectrum of these operators are known.
In the paper under consideration the authors discuss the problem of stochastic stability of the isolated eigenvalues in a general framework for bounded linear operators acting in an abstract Banach space and satisfying the so called Lasota-Yorke type inequality well known in the theory of piecewise expanding maps.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37A50 Dynamical systems and their relations with probability theory and stochastic processes
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37H99 Random dynamical systems
47A35 Ergodic theory of linear operators
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References:

[1] V. Baladi - M. Holschneider , Approximation of nonessential spectrum of transfer operators , Preprint ( 1998 ). MR 1690191 · Zbl 0984.47002
[2] V. Baladi - L.S. Young , On the spectra of randomly perturbed expanding maps , Comm. Math. Phys. 156 ( 1993 ), 355 - 385 ; 166 ( 1994 ), 219 - 220 . MR 1233850 | Zbl 0809.60101 · Zbl 0809.60101 · doi:10.1007/BF02098487
[3] M.L. Blank , Stochastic properties of deterministic dynamical systems , Soviet Sci. Rev. Sect. C Math. Phys. Rev. 6 ( 1987 ), 243 - 271 . MR 968674 | Zbl 0634.58019 · Zbl 0634.58019
[4] M.L. Blank , Small perturbations of chaotic dynamical systems , Uspekhi Mat. Nauk 44 ( 1989 ), 3 - 28 . MR 1037009 | Zbl 0702.58063 · Zbl 0702.58063 · doi:10.1070/RM1989v044n06ABEH002302
[5] M.L. Blank - G. Keller , Stochastic stability versus localization in chaotic dynamical systems , Nonlinearity 10 ( 1997 ), 81 - 107 . MR 1430741 | Zbl 0907.58011 · Zbl 0907.58011 · doi:10.1088/0951-7715/10/1/006
[6] M.L. Blank - G. Keller , Random perturbations of chaotic dynamical systems: Stability of the spectrum , Preprint ( 1998 ). MR 1644405 · Zbl 0921.58051
[7] A. Boyarsky - P. Góra , Absolutely continuous invariant measures for piecewise expanding C2 transformations in RN , Israel J. Math. 67 ( 1989 ), 272 - 286 . MR 1029902 | Zbl 0691.28004 · Zbl 0691.28004 · doi:10.1007/BF02764946
[8] A. Boyarsky - P. Góra , ” Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension ”, Birkhäuser , Boston , 1997 . MR 1461536 | Zbl 0893.28013 · Zbl 0893.28013
[9] C. Chiu - Q. Du - T.Y. Li , Error estimates of the Markov finite approximation to the Frobenius-Perron operator , Nonlinear Anal. 19 ( 1992 ), 291 - 308 . MR 1178404 | Zbl 0788.28009 · Zbl 0788.28009 · doi:10.1016/0362-546X(92)90175-E
[10] N. Dunford - J.T. Schwartz , ” Linear Operators, Part I: General Theory ”, Wiley , 1957 . MR 1009162 | Zbl 0635.47001 · Zbl 0635.47001
[11] G. Froyland , Computer-assisted bounds for the rate of decay of correlations , Comm. Math. Phys. 189 ( 1997 ), 237 - 257 . MR 1478538 | Zbl 0892.58047 · Zbl 0892.58047 · doi:10.1007/s002200050198
[12] H. Hennion , Sur un théorème spectral et son application aux noyaux Lipchitziens , Proc. Amer. Math. Soc. 118 ( 1993 ), 627 - 634 . MR 1129880 | Zbl 0772.60049 · Zbl 0772.60049 · doi:10.2307/2160348
[13] F. Hofbauer - G. Keller , Ergodic properties of invariant measures for piecewise monotonic transformations , Math. Z. 180 ( 1982 ), 119 - 140 . Article | MR 656227 | Zbl 0485.28016 · Zbl 0485.28016 · doi:10.1007/BF01215004
[14] F.Y. Hunt - W. Miller , On the approximation of invariant measures , J. Statist. Phys. 66 ( 1992 ), 535 - 548 . MR 1149495 | Zbl 0892.58048 · Zbl 0892.58048 · doi:10.1007/BF01060079
[15] C.T. Ionescu Tulcea - G. Marinescu , Thórie ergodique pour des classes d’opérations non complètement continues , Ann. of Math. 52 ( 1950 ), 140 - 147 . MR 37469 | Zbl 0040.06502 · Zbl 0040.06502 · doi:10.2307/1969514
[16] M. Iosifescu - R. Theodorescu , ” Random Processes and Learning ”, Grundlehren Math. Wiss. , Vol. 150 , Springer , 1969 . MR 293704 | Zbl 0194.51101 · Zbl 0194.51101
[17] M. Keane - R. Murray - L.S. Young , Computing invariant measures for expanding circle maps , Nonlinearity 11 ( 1998 ), 27 - 46 . MR 1492949 | Zbl 0903.58019 · Zbl 0903.58019 · doi:10.1088/0951-7715/11/1/004
[18] G. Keller , Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d’une région bornée du plan , C.R.Acad. Sci. Paris , Série A 289 ( 1979 ), 625 - 627 (Kurzfassung der Dissertation). MR 556443 | Zbl 0419.28007 · Zbl 0419.28007
[19] G. Keller , Un théorème de la limite centrale pour une classe de transformations monotones par morceaux , C. R. Acad. Sci. Paris , Série A , 291 ( 1980 ), 155 - 158 . MR 605005 | Zbl 0446.60013 · Zbl 0446.60013
[20] G. Keller , On the rate of convergence to equilibrium in one-dimensional systems , Comm. Math. Phys. 96 ( 1984 ), 181 - 193 . Article | MR 768254 | Zbl 0576.58016 · Zbl 0576.58016 · doi:10.1007/BF01240219
[21] G. Keller , Stochastic stability in some chaotic dynamical systems , Monatsh. Math. 94 ( 1982 ), 313 - 333 . MR 685377 | Zbl 0496.58010 · Zbl 0496.58010 · doi:10.1007/BF01667385
[22] G. Keller - M. Künzle , Transfer operators for coupled map lattices , Ergodic Theory Dynam. Systems 12 ( 1992 ), 297 - 318 . MR 1176625 | Zbl 0737.58032 · Zbl 0737.58032 · doi:10.1017/S0143385700006763
[23] G. Keller - T. Nowicki , Spectral theory, zeta functions and the distribution of periodic orbits for Collet-Eckmann maps , Comm. Math. Phys. 149 ( 1992 ), 31 - 69 . Article | MR 1182410 | Zbl 0763.58024 · Zbl 0763.58024 · doi:10.1007/BF02096623
[24] A. Lasota - J.A. Yorke , On the existence of invariant measures for piecewise monotonic transformations , Trans. Amer. Math. Soc. 186 ( 1973 ), 481 - 488 . MR 335758 | Zbl 0298.28015 · Zbl 0298.28015 · doi:10.2307/1996575
[25] T.Y. Li , Finite approximations for the Frobenius-Perron operator: A solution to Ulam’s conjecture , J. Approx. Theory 17 ( 1976 ), 177 - 186 . MR 412689 | Zbl 0357.41011 · Zbl 0357.41011 · doi:10.1016/0021-9045(76)90037-X
[26] W. Miller , Stability and approximation of invariant measures for a class of nonexpanding transformations , Nonlinear Anal. 23 ( 1994 ), 1013 - 1025 . MR 1304241 | Zbl 0822.28009 · Zbl 0822.28009 · doi:10.1016/0362-546X(94)90196-1
[27] M.F. Norman , ” Markov Processes and Learning Models ”, Mathematics in Science and Engineering , Vol. 84 , Academic Press , 1972 . MR 423546 | Zbl 0262.92003 · Zbl 0262.92003
[28] W. Parry - M. Pollicott , Zeta functions and the periodic orbit structure of hyperbolic dynamics , Astérisque , Vol. 187 - 188 , 1990 . MR 1085356 | Zbl 0726.58003 · Zbl 0726.58003
[29] A. Pinkus , ” n-Widths in Approximation Theory ”, Springer , 1985 . MR 774404 | Zbl 0551.41001 · Zbl 0551.41001
[30] M. Rychlik , Bounded variation and invariant measures , Studia Math. 76 ( 1983 ), 69 - 80 . Article | MR 728198 | Zbl 0575.28011 · Zbl 0575.28011
[31] H.H. Schaefer , ” Banach Lattices and Positive Operators ”, Grundlehren Math. Wiss. , Vol. 215 , Springer , 1974 . MR 423039 | Zbl 0296.47023 · Zbl 0296.47023
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