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Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space
Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps
1. | Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56100 Pisa, Italy |
2. | Instituto de Matemática, Av. Lourival Melo Mota, Tabuleiro do Martins, s/n, Maceió -AL, CEP 57072-900, Brazil |
We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation).
As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $ \delta $, the physical measure varies continuously, with a modulus of continuity $ O(\delta \log \delta ) $, which is asymptotically optimal for this kind of piecewise smooth maps.
References:
[1] |
J. F. Alves and M. Soufi,
Statistical stability of geometric Lorenz attractors, Fund. Math., 224 (2014), 219-231.
doi: 10.4064/fm224-3-2. |
[2] |
V. Araujo, S. Galatolo and M. Pacifico,
Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Math. Z., 276 (2014), 1001-1048.
doi: 10.1007/s00209-013-1231-0. |
[3] |
V. Araujo and M. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53, Springer, Heidelberg, 2010.
doi: 10.1007/978-3-642-11414-4. |
[4] |
W. Bahsoun and M. Ruziboev,
On the statistical stability of Lorenz attractors with a $C^{1+\alpha}$ stable foliation, Ergodic Theory Dynam. Systems, 39 (2019), 3169-3184.
doi: 10.1017/etds.2018.28. |
[5] |
V. Baladi,
The quest for the ultimate anisotropic Banach space, J. Stat. Phys., 166 (2017), 525-557.
doi: 10.1007/s10955-016-1663-0. |
[6] |
V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812813633. |
[7] |
V. Baladi and M. Tsujii,
Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.
doi: 10.5802/aif.2253. |
[8] |
V. Baladi and S. Gouëzel,
Banach spaces for piecewise cone-hyperbolic maps, J. Mod. Dyn., 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[9] |
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. |
[10] |
A. Boyarsky and P. Gora, Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications, Birkhäuser, Boston, MA, 1997.
doi: 10.1007/978-1-4612-2024-4. |
[11] |
O. Butterley and C. Liverani,
Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322.
doi: 10.3934/jmd.2007.1.301. |
[12] |
O. Butterley and I. Melbourne,
Disintegration of invariant measures for hyperbolic skew products, Israel J. Math., 219 (2017), 171-188.
doi: 10.1007/s11856-017-1477-z. |
[13] |
M. Demers,
A gentle introduction to anisotropic Banach spaces, Chaos Solitons Fractals, 116 (2018), 29-42.
doi: 10.1016/j.chaos.2018.08.028. |
[14] |
M. 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. |
[15] |
M. Demers and H. Z. Zhang,
Spectral analysis of the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.
doi: 10.3934/jmd.2011.5.665. |
[16] |
M. Demers and H. Z. Zhang,
A functional analytic approach to perturbations of the Lorentz gas, CComm. Math. Phys., 324 (2013), 767-830.
doi: 10.1007/s00220-013-1820-0. |
[17] |
S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: math/1510.02615. |
[18] |
S. Galatolo, Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products, J. Éc. polytech. Math., 5 (2018), 377–405.
doi: 10.5802/jep.73. |
[19] |
S. Galatolo, I. Nisoli and B. Saussol,
An elementary way to rigorously estimate convergence to equilibrium and escape rates, J. Comput. Dyn., 2 (2015), 51-64.
doi: 10.3934/jcd.2015.2.51. |
[20] |
S. Galatolo and M. J. Pacifico,
Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence, Ergodic Theory Dynam. Systems, 30 (2010), 1703-1737.
doi: 10.1017/S0143385709000856. |
[21] |
S. Gouezel and C. Liverani,
Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.
doi: 10.1017/S0143385705000374. |
[22] |
C. Ionescu-Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérateurs non complètement continues, Ann. of Math. (2), 52 (1950), 140–147.
doi: 10.2307/1969514. |
[23] |
G. Keller,
Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.
doi: 10.1007/BF00532744. |
[24] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141–152. |
[25] |
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. |
[26] |
C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems, Part II, Scuola Norm. Sup., Pisa, 2003, 185–237. |
[27] |
C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239–301.
doi: 10.2307/2118636. |
[28] |
R. Lucena, Spectral Gap for Contracting Fiber Systems and Applications, Ph.D thesis, Universidade Federal do Rio de Janeiro in Brazil, 2015. |
[29] |
K. Oliveira and M. Viana, Fundamentos da Teoria Ergódica, Colecão Fronteiras da Matematica - SBM, Brazil, 2014. |
[30] |
J. Rousseau-Egele,
Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.
doi: 10.1214/aop/1176993522. |
[31] |
M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium, IMPA, 1997, IMPA. Availble from: http://w3.impa.br/viana/out/sdds.pdf. |
show all references
References:
[1] |
J. F. Alves and M. Soufi,
Statistical stability of geometric Lorenz attractors, Fund. Math., 224 (2014), 219-231.
doi: 10.4064/fm224-3-2. |
[2] |
V. Araujo, S. Galatolo and M. Pacifico,
Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Math. Z., 276 (2014), 1001-1048.
doi: 10.1007/s00209-013-1231-0. |
[3] |
V. Araujo and M. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53, Springer, Heidelberg, 2010.
doi: 10.1007/978-3-642-11414-4. |
[4] |
W. Bahsoun and M. Ruziboev,
On the statistical stability of Lorenz attractors with a $C^{1+\alpha}$ stable foliation, Ergodic Theory Dynam. Systems, 39 (2019), 3169-3184.
doi: 10.1017/etds.2018.28. |
[5] |
V. Baladi,
The quest for the ultimate anisotropic Banach space, J. Stat. Phys., 166 (2017), 525-557.
doi: 10.1007/s10955-016-1663-0. |
[6] |
V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812813633. |
[7] |
V. Baladi and M. Tsujii,
Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.
doi: 10.5802/aif.2253. |
[8] |
V. Baladi and S. Gouëzel,
Banach spaces for piecewise cone-hyperbolic maps, J. Mod. Dyn., 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[9] |
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. |
[10] |
A. Boyarsky and P. Gora, Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications, Birkhäuser, Boston, MA, 1997.
doi: 10.1007/978-1-4612-2024-4. |
[11] |
O. Butterley and C. Liverani,
Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322.
doi: 10.3934/jmd.2007.1.301. |
[12] |
O. Butterley and I. Melbourne,
Disintegration of invariant measures for hyperbolic skew products, Israel J. Math., 219 (2017), 171-188.
doi: 10.1007/s11856-017-1477-z. |
[13] |
M. Demers,
A gentle introduction to anisotropic Banach spaces, Chaos Solitons Fractals, 116 (2018), 29-42.
doi: 10.1016/j.chaos.2018.08.028. |
[14] |
M. 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. |
[15] |
M. Demers and H. Z. Zhang,
Spectral analysis of the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.
doi: 10.3934/jmd.2011.5.665. |
[16] |
M. Demers and H. Z. Zhang,
A functional analytic approach to perturbations of the Lorentz gas, CComm. Math. Phys., 324 (2013), 767-830.
doi: 10.1007/s00220-013-1820-0. |
[17] |
S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: math/1510.02615. |
[18] |
S. Galatolo, Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products, J. Éc. polytech. Math., 5 (2018), 377–405.
doi: 10.5802/jep.73. |
[19] |
S. Galatolo, I. Nisoli and B. Saussol,
An elementary way to rigorously estimate convergence to equilibrium and escape rates, J. Comput. Dyn., 2 (2015), 51-64.
doi: 10.3934/jcd.2015.2.51. |
[20] |
S. Galatolo and M. J. Pacifico,
Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence, Ergodic Theory Dynam. Systems, 30 (2010), 1703-1737.
doi: 10.1017/S0143385709000856. |
[21] |
S. Gouezel and C. Liverani,
Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.
doi: 10.1017/S0143385705000374. |
[22] |
C. Ionescu-Tulcea and G. Marinescu, Théorie ergodique pour des classes d'opérateurs non complètement continues, Ann. of Math. (2), 52 (1950), 140–147.
doi: 10.2307/1969514. |
[23] |
G. Keller,
Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.
doi: 10.1007/BF00532744. |
[24] |
G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141–152. |
[25] |
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. |
[26] |
C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems, Part II, Scuola Norm. Sup., Pisa, 2003, 185–237. |
[27] |
C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239–301.
doi: 10.2307/2118636. |
[28] |
R. Lucena, Spectral Gap for Contracting Fiber Systems and Applications, Ph.D thesis, Universidade Federal do Rio de Janeiro in Brazil, 2015. |
[29] |
K. Oliveira and M. Viana, Fundamentos da Teoria Ergódica, Colecão Fronteiras da Matematica - SBM, Brazil, 2014. |
[30] |
J. Rousseau-Egele,
Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab., 11 (1983), 772-788.
doi: 10.1214/aop/1176993522. |
[31] |
M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium, IMPA, 1997, IMPA. Availble from: http://w3.impa.br/viana/out/sdds.pdf. |
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