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From log Sobolev to Talagrand: A quick proof
Stochastic perturbations and Ulam's method for W-shaped maps
1. | Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada, Canada |
References:
[1] |
Ch. J. Bose and R. Murray, The exact rate of approximation in Ulam's method, Discrete and Continuous Dynamical Systems, 7 (2001), 219-235. |
[2] |
A. Boyarsky and P. Góra, "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. |
[3] |
Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems," Tsinghua University Texts, 2009.
doi: 10.1007/978-3-540-85367-1. |
[4] |
P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations, Preprint, available from: http://www.mathstat.concordia.ca/faculty/pgora/EslamiGora_Stronger_LY_inequality_rev3.pdf. |
[5] |
P. Eslami and M. Misiurewicz, Singular limits of absolutely continuous invariant measures for families of transitive map, Journal of Difference Equations and Applications.
doi: 10.1080/10236198.2011.590480. |
[6] |
P. Góra, On small stochastic perturbations of mappings of the unit interval, Colloq. Math., 49 (1984), 73-85. |
[7] |
G. Keller, Stochastic stability in some chaotic dynamical systems, Monatshefte für Mathematik, 94 (1982), 313-333.
doi: 10.1007/BF01667385. |
[8] |
G. Keller and C. Liverani., Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152. |
[9] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. |
[10] |
Z. Li, P. Góra, A. Boyarsky, H. Proppe and P. Eslami, A family of piecewise expanding maps having singular measure as a limit of acim's, in press, Ergodic Theory and Dynamical Systems.
doi: 10.1017/S0143385711000836. |
[11] |
Z. Li, W-like maps with various instabilities of acim's, Preprint, arXiv:1109.5199. |
[12] |
T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture, Jour. Approx. Theory, 17 (1976), 177-186. |
[13] |
R. Murray, Ulam's method for some non-uniformly expanding maps, Discrete and Continuous Dynamical Systems, 26 (2010), 1007-1018.
doi: 10.3934/dcds.2010.26.1007. |
[14] |
R. Murray, Existence, mixing and approximation of invariant densities for expanding maps on $R^r$, Nonlinear Analysis TMA, 45 (2001), 37-72.
doi: 10.1016/S0362-546X(99)00329-6. |
[15] |
S. M. Ulam, "A Collection of Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. |
show all references
References:
[1] |
Ch. J. Bose and R. Murray, The exact rate of approximation in Ulam's method, Discrete and Continuous Dynamical Systems, 7 (2001), 219-235. |
[2] |
A. Boyarsky and P. Góra, "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. |
[3] |
Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems," Tsinghua University Texts, 2009.
doi: 10.1007/978-3-540-85367-1. |
[4] |
P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations, Preprint, available from: http://www.mathstat.concordia.ca/faculty/pgora/EslamiGora_Stronger_LY_inequality_rev3.pdf. |
[5] |
P. Eslami and M. Misiurewicz, Singular limits of absolutely continuous invariant measures for families of transitive map, Journal of Difference Equations and Applications.
doi: 10.1080/10236198.2011.590480. |
[6] |
P. Góra, On small stochastic perturbations of mappings of the unit interval, Colloq. Math., 49 (1984), 73-85. |
[7] |
G. Keller, Stochastic stability in some chaotic dynamical systems, Monatshefte für Mathematik, 94 (1982), 313-333.
doi: 10.1007/BF01667385. |
[8] |
G. Keller and C. Liverani., Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152. |
[9] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. |
[10] |
Z. Li, P. Góra, A. Boyarsky, H. Proppe and P. Eslami, A family of piecewise expanding maps having singular measure as a limit of acim's, in press, Ergodic Theory and Dynamical Systems.
doi: 10.1017/S0143385711000836. |
[11] |
Z. Li, W-like maps with various instabilities of acim's, Preprint, arXiv:1109.5199. |
[12] |
T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture, Jour. Approx. Theory, 17 (1976), 177-186. |
[13] |
R. Murray, Ulam's method for some non-uniformly expanding maps, Discrete and Continuous Dynamical Systems, 26 (2010), 1007-1018.
doi: 10.3934/dcds.2010.26.1007. |
[14] |
R. Murray, Existence, mixing and approximation of invariant densities for expanding maps on $R^r$, Nonlinear Analysis TMA, 45 (2001), 37-72.
doi: 10.1016/S0362-546X(99)00329-6. |
[15] |
S. M. Ulam, "A Collection of Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. |
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