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Stochastic perturbations and Ulam's method for Wshaped 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. 
[2] 
A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997). doi: 10.1007/9781461220244. 
[3] 
Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009). doi: 10.1007/9783540853671. 
[4] 
P. Eslami and P. Góra, Stronger LasotaYorke inequality for piecewise monotonic transformations,, Preprint, (). 
[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. 
[7] 
G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313. 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. 
[9] 
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. 
[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, (). doi: 10.1017/S0143385711000836. 
[11] 
Z. Li, Wlike maps with various instabilities of acim's,, Preprint, (). 
[12] 
T. Y. Li, Finite approximation for the FrobeniusPerron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177. 
[13] 
R. Murray, Ulam's method for some nonuniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007. 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. doi: 10.1016/S0362546X(99)003296. 
[15] 
S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (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. 
[2] 
A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997). doi: 10.1007/9781461220244. 
[3] 
Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009). doi: 10.1007/9783540853671. 
[4] 
P. Eslami and P. Góra, Stronger LasotaYorke inequality for piecewise monotonic transformations,, Preprint, (). 
[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. 
[7] 
G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313. 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. 
[9] 
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. 
[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, (). doi: 10.1017/S0143385711000836. 
[11] 
Z. Li, Wlike maps with various instabilities of acim's,, Preprint, (). 
[12] 
T. Y. Li, Finite approximation for the FrobeniusPerron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177. 
[13] 
R. Murray, Ulam's method for some nonuniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007. 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. doi: 10.1016/S0362546X(99)003296. 
[15] 
S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960). 
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