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September  2019, 39(9): 4955-4977. doi: 10.3934/dcds.2019203

Statistical properties of one-dimensional expanding maps with singularities of low regularity

Jianyu Chen and  Hong-Kun Zhang

Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA

Received  March 2018 Revised  January 2019 Published  May 2019

Fund Project: The second author is partially supported by the NSF Career Award (DMS-1151762).

We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. The method is new yet simple: instead of directly working with the 1-d map, we first lift the 1-d expanding map to a hyperbolic map on the unit square, and then take advantage of the functional analytic method developed by Demers and Zhang in [21,22,23] for hyperbolic systems with singularities. By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables. Moreover, we are able to prove that the projected SRB measure has a piecewise continuous density function. Our results apply to rather general 1-d expanding maps, including some $ C^1 $ perturbations of the Lorenz-like map and the Gauss map whose statistical properties are still unknown as they fail all other available methods.

Keywords: Expanding maps, singularities, statistical properties.
Mathematics Subject Classification: Primary: 37D50, 37A25.
Citation: Jianyu Chen, Hong-Kun Zhang. Statistical properties of one-dimensional expanding maps with singularities of low regularity. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 4955-4977. doi: 10.3934/dcds.2019203
References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, volume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, Algebraic and Topological Dynamics, volume 385 of Contemp. Math., pages 123–135. Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/385/07194.  Google Scholar

[3]

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

[4]

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

[5]

V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys., 127 (1990), 459-477.  doi: 10.1007/BF02104498.  Google Scholar

[6]

V. Baladi and C. Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Comm. Math. Phys., 314 (2012), 689-773.  doi: 10.1007/s00220-012-1538-4.  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 M. Tsujii, Spectra of differentiable hyperbolic maps, Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., E38, pages 1–21. Friedr. Vieweg, Wiesbaden, 2008.  Google Scholar

[9]

V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.  doi: 10.1007/BF02098487.  Google Scholar

[10]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  doi: 10.1088/0951-7715/15/6/309.  Google Scholar

[11]

F. Bonetto, N. Chernov, A. Korepanov and J.-L. Lebowitz, Spatial structure of stationary nonequilibrium states in the thermostatted periodic Lorentz gas, J. Stat. Phys., 146 (2012), 1221-1243.  doi: 10.1007/s10955-012-0444-7.  Google Scholar

[12]

R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1–17. With an afterword by Roy L. Adler and additional comments by Caroline Series. doi: 10.1007/BF01941319.  Google Scholar

[13]

L. A. Bunimovich, Ya. G. Sinai and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Uspekhi Mat. Nauk, 46 (1991), 43–92,192. doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar

[14]

O. Butterley, An alternative approach to generalised BV and the application to expanding interval maps, Discrete Contin. Dyn. Syst., 33 (2013), 3355-3363.  doi: 10.3934/dcds.2013.33.3355.  Google Scholar

[15]

O. Butterley, Area expanding ${C}^{1+\alpha}$ suspension semiflows, Comm. Math. Phys., 325 (2014), 803-820.  doi: 10.1007/s00220-013-1835-6.  Google Scholar

[16]

N. Chernov and A. Korepanov, Spatial structure of Sinai-Ruelle-Bowen measures, Phys. D, 285 (2014), 1-7.  doi: 10.1016/j.physd.2014.06.006.  Google Scholar

[17]

N. Chernov and R. Markarian, Chaotic Billiards, Volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar

[18]

N. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3.  Google Scholar

[19]

W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory Dynam. Systems, 22 (2002), 1061-1078.  doi: 10.1017/S0143385702000627.  Google Scholar

[20]

M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 92008), 4777–4814. doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[21]

M. Demers and H.-K. 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

[22]

M. Demers and H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.  doi: 10.1007/s00220-013-1820-0.  Google Scholar

[23]

M. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379.  Google Scholar

[24]

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

[25]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties, J. Differential Geom., 79 (2008), 433-477.  doi: 10.4310/jdg/1213798184.  Google Scholar

[26]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.  doi: 10.1007/BF01215004.  Google Scholar

[27]

H. Hu and S. Vaienti, Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 29 (2009), 1185-1215.  doi: 10.1017/S0143385708000576.  Google Scholar

[28]

H. Hu and S. Vaienti, Lower bounds for the decay of correlations in non-uniformly expanding maps, Ergodic Theory and Dynamical Systems, 2017, 1–35. doi: 10.1017/etds.2017.107.  Google Scholar

[29]

A. Katok, J.-M. Strelcyn, F. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0099031.  Google Scholar

[30]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.  doi: 10.1007/BF01240219.  Google Scholar

[31]

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

[32]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481–488 (1974). doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar

[33]

T. Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192.  doi: 10.1090/S0002-9947-1978-0457679-0.  Google Scholar

[34]

C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[35]

C. Liverani, A footnote on expanding maps, Discrete Contin. Dyn. Syst., 33 (2013), 3741-3751.  doi: 10.3934/dcds.2013.33.3741.  Google Scholar

[36]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory Dynam. Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.  Google Scholar

[37]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  doi: 10.4064/sm-76-1-69-80.  Google Scholar

[38]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar

[39]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

[40]

M. Tsujii, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495-1545.  doi: 10.1088/0951-7715/23/7/001.  Google Scholar

[41]

M. Viana, Lecture Notes on Attractors and Physical Measures, volume 8 of Monografías del Instituto de Matemática y Ciencias Afines [Monographs of the Institute of Mathematics and Related Sciences]. Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. A paper from the 12th Escuela Latinoamericana de Matemáticas (Ⅻ-ELAM) held in Lima, June 28–July 3, 1999.  Google Scholar

[42]

S. Wong, Hölder continuous derivatives and ergodic theory, J. London Math. Soc. (2), 22 (1980), 506-520.  doi: 10.1112/jlms/s2-22.3.506.  Google Scholar

show all references

References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, volume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ foliations, Algebraic and Topological Dynamics, volume 385 of Contemp. Math., pages 123–135. Amer. Math. Soc., Providence, RI, 2005. doi: 10.1090/conm/385/07194.  Google Scholar

[3]

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

[4]

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

[5]

V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys., 127 (1990), 459-477.  doi: 10.1007/BF02104498.  Google Scholar

[6]

V. Baladi and C. Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Comm. Math. Phys., 314 (2012), 689-773.  doi: 10.1007/s00220-012-1538-4.  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 M. Tsujii, Spectra of differentiable hyperbolic maps, Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., E38, pages 1–21. Friedr. Vieweg, Wiesbaden, 2008.  Google Scholar

[9]

V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385.  doi: 10.1007/BF02098487.  Google Scholar

[10]

M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  doi: 10.1088/0951-7715/15/6/309.  Google Scholar

[11]

F. Bonetto, N. Chernov, A. Korepanov and J.-L. Lebowitz, Spatial structure of stationary nonequilibrium states in the thermostatted periodic Lorentz gas, J. Stat. Phys., 146 (2012), 1221-1243.  doi: 10.1007/s10955-012-0444-7.  Google Scholar

[12]

R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1–17. With an afterword by Roy L. Adler and additional comments by Caroline Series. doi: 10.1007/BF01941319.  Google Scholar

[13]

L. A. Bunimovich, Ya. G. Sinai and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Uspekhi Mat. Nauk, 46 (1991), 43–92,192. doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar

[14]

O. Butterley, An alternative approach to generalised BV and the application to expanding interval maps, Discrete Contin. Dyn. Syst., 33 (2013), 3355-3363.  doi: 10.3934/dcds.2013.33.3355.  Google Scholar

[15]

O. Butterley, Area expanding ${C}^{1+\alpha}$ suspension semiflows, Comm. Math. Phys., 325 (2014), 803-820.  doi: 10.1007/s00220-013-1835-6.  Google Scholar

[16]

N. Chernov and A. Korepanov, Spatial structure of Sinai-Ruelle-Bowen measures, Phys. D, 285 (2014), 1-7.  doi: 10.1016/j.physd.2014.06.006.  Google Scholar

[17]

N. Chernov and R. Markarian, Chaotic Billiards, Volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar

[18]

N. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3.  Google Scholar

[19]

W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory Dynam. Systems, 22 (2002), 1061-1078.  doi: 10.1017/S0143385702000627.  Google Scholar

[20]

M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 92008), 4777–4814. doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[21]

M. Demers and H.-K. 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

[22]

M. Demers and H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.  doi: 10.1007/s00220-013-1820-0.  Google Scholar

[23]

M. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379.  Google Scholar

[24]

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

[25]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties, J. Differential Geom., 79 (2008), 433-477.  doi: 10.4310/jdg/1213798184.  Google Scholar

[26]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.  doi: 10.1007/BF01215004.  Google Scholar

[27]

H. Hu and S. Vaienti, Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 29 (2009), 1185-1215.  doi: 10.1017/S0143385708000576.  Google Scholar

[28]

H. Hu and S. Vaienti, Lower bounds for the decay of correlations in non-uniformly expanding maps, Ergodic Theory and Dynamical Systems, 2017, 1–35. doi: 10.1017/etds.2017.107.  Google Scholar

[29]

A. Katok, J.-M. Strelcyn, F. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0099031.  Google Scholar

[30]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.  doi: 10.1007/BF01240219.  Google Scholar

[31]

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

[32]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481–488 (1974). doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar

[33]

T. Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192.  doi: 10.1090/S0002-9947-1978-0457679-0.  Google Scholar

[34]

C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[35]

C. Liverani, A footnote on expanding maps, Discrete Contin. Dyn. Syst., 33 (2013), 3741-3751.  doi: 10.3934/dcds.2013.33.3741.  Google Scholar

[36]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory Dynam. Systems, 33 (2013), 168-182.  doi: 10.1017/S0143385711000939.  Google Scholar

[37]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  doi: 10.4064/sm-76-1-69-80.  Google Scholar

[38]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar

[39]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917.  Google Scholar

[40]

M. Tsujii, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495-1545.  doi: 10.1088/0951-7715/23/7/001.  Google Scholar

[41]

M. Viana, Lecture Notes on Attractors and Physical Measures, volume 8 of Monografías del Instituto de Matemática y Ciencias Afines [Monographs of the Institute of Mathematics and Related Sciences]. Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999. A paper from the 12th Escuela Latinoamericana de Matemáticas (Ⅻ-ELAM) held in Lima, June 28–July 3, 1999.  Google Scholar

[42]

S. Wong, Hölder continuous derivatives and ergodic theory, J. London Math. Soc. (2), 22 (1980), 506-520.  doi: 10.1112/jlms/s2-22.3.506.  Google Scholar

Figure 1.  Lorenz-like Map
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Figure 2.  Gauss Map
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Figure 3.  The two dimensional lifting map $ \widehat{F}: Q\backslash \widehat{\mathcal{S}}_1\to Q\backslash \widehat{\mathcal{S}}_{-1} $
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