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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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.

[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.

[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.

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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