Stochastic perturbations and Ulam's method for W-shaped maps

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May 2013, 33(5): 1937-1944. doi: 10.3934/dcds.2013.33.1937

Stochastic perturbations and Ulam's method for W-shaped maps

Paweł Góra ^1, and Abraham Boyarsky ^1,

1.	Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada, Canada

Received November 2011 Revised January 2012 Published December 2012

Abstract
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For a discrete dynamical system given by a map $\tau :I\rightarrow I$, the long term behavior is described by the probability density function (pdf) of an absolutely continuous invariant measure. This pdf is the fixed point of the Frobenius-Perron operator on $L^{1}(I)$ induced by $\tau$. Ulam suggested a numerical procedure for approximating a pdf by using matrix approximations to the Frobenius-Perron operator. In [12] Li proved the convergence for maps which are piecewise $C^{2}$ and satisfy $| \tau'| >2.$ In this paper we will consider a larger class of maps with weaker smoothness conditions and a harmonic slope condition which permits slopes equal to $\pm $2. Using a generalized Lasota-Yorke inequality [4], we establish convergence for the Ulam approximation method for this larger class of maps. Ulam's method is a special case of small stochastic perturbations. We obtain stability of the pdf under such perturbations. Although our conditions apply to many maps, there are important examples which do not satisfy these conditions, for example the $W$-map [7]. The $W$-map is highly unstable in the sense that it is possible to construct perturbations $W_a$ with absolutely continuous invariant measures (acim) $\mu_a$ such that $\mu_a$ converge to a singular measure although $W_a$ converge to $W$. We prove the convergence of Ulam's method for the $W$-map by direct calculations.

Keywords: Markov maps, Frobenius-Perron operator, Piecewise expanding maps of an interval, harmonic average of slopes., Ulam's method, W-shaped maps, absolutely continuous invariant measures.

Mathematics Subject Classification: Primary: 37A10; Secondary: 37A05, 37E0.

Citation: Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937

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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