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March  2020, 40(3): 1309-1360. doi: 10.3934/dcds.2020079

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

Received  November 2018 Revised  August 2019 Published  December 2019

Fund Project: This work was partially supported by Alagoas Research Foundation-FAPEAL (Brazil) Grants 60030 000587/2016, CNPq (Brazil) Grants 300398/2016-6, CAPES (Brazil) Grants 99999.014021/2013-07 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE- 2012-IRSES 318999 BREUDS).

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.

Citation: Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1309-1360. doi: 10.3934/dcds.2020079
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.  Google Scholar

[2]

V. AraujoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: math/1510.02615. Google Scholar

[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.  Google Scholar

[19]

S. GalatoloI. 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.  Google Scholar

[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.  Google Scholar

[21]

S. Gouezel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[23]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.  doi: 10.1007/BF00532744.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239–301. doi: 10.2307/2118636.  Google Scholar

[28]

R. Lucena, Spectral Gap for Contracting Fiber Systems and Applications, Ph.D thesis, Universidade Federal do Rio de Janeiro in Brazil, 2015. Google Scholar

[29]

K. Oliveira and M. Viana, Fundamentos da Teoria Ergódica, Colecão Fronteiras da Matematica - SBM, Brazil, 2014. Google Scholar

[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.  Google Scholar

[31]

M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium, IMPA, 1997, IMPA. Availble from: http://w3.impa.br/viana/out/sdds.pdf. Google Scholar

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.  Google Scholar

[2]

V. AraujoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: math/1510.02615. Google Scholar

[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.  Google Scholar

[19]

S. GalatoloI. 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.  Google Scholar

[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.  Google Scholar

[21]

S. Gouezel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[23]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.  doi: 10.1007/BF00532744.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239–301. doi: 10.2307/2118636.  Google Scholar

[28]

R. Lucena, Spectral Gap for Contracting Fiber Systems and Applications, Ph.D thesis, Universidade Federal do Rio de Janeiro in Brazil, 2015. Google Scholar

[29]

K. Oliveira and M. Viana, Fundamentos da Teoria Ergódica, Colecão Fronteiras da Matematica - SBM, Brazil, 2014. Google Scholar

[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.  Google Scholar

[31]

M. Viana, Stochastic dynamics of deterministic systems, Brazillian Math. Colloquium, IMPA, 1997, IMPA. Availble from: http://w3.impa.br/viana/out/sdds.pdf. Google Scholar

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