Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Previous Article
Positive Lyapunov exponent for a class of quasi-periodic cocycles
DCDS Home
This Issue
Next Article
Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space

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 and 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.
[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.

[1]	Youngna Choi. Attractors from one dimensional Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 715-730. doi: 10.3934/dcds.2004.11.715
[2]	Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[3]	Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159
[4]	Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112
[5]	Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[6]	Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. Mathematical Biosciences & Engineering, 2004, 1 (1) : 161-184. doi: 10.3934/mbe.2004.1.161
[7]	Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[8]	Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
[9]	O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[10]	Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2187-2209. doi: 10.3934/cpaa.2021063
[11]	Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4297-4318. doi: 10.3934/dcds.2021037
[12]	José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[13]	Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[14]	Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[15]	Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005
[16]	Shuang Chen, Jun Shen. Large spectral gap induced by small delay and its application to reduction. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190
[17]	Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic and Related Models, 2022, 15 (1) : 91-117. doi: 10.3934/krm.2021045
[18]	Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[19]	Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[20]	Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Stefano Galatolo Rafael Lucena
on Google Scholar
Stefano Galatolo Rafael Lucena

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Stefano Galatolo Rafael Lucena
on Google Scholar
Stefano Galatolo Rafael Lucena

[Back to Top]