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Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions
Statistical properties of one-dimensional expanding maps with singularities of low regularity
Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA |
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 [
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. |



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