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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

Stefano Galatolo 1, and  Rafael Lucena 2,

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.

Keywords: Spectral gap, statistical properties, Lorenz-Like, transfer operator, stability.
Mathematics Subject Classification: Primary: 37A25, 37A10; Secondary: 37C30, 37D50.
Citation: Stefano Galatolo, Rafael Lucena. Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete & Continuous Dynamical Systems, 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. 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.  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. 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.  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. 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.  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. 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.  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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