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

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[2]	A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997). doi: 10.1007/978-1-4612-2024-4. Google Scholar
[3]	Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009). doi: 10.1007/978-3-540-85367-1. Google Scholar
[4]	P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations,, Preprint, (). Google Scholar
[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. Google Scholar
[6]	P. Góra, On small stochastic perturbations of mappings of the unit interval,, Colloq. Math., 49 (1984), 73. Google Scholar
[7]	G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313. doi: 10.1007/BF01667385. Google Scholar
[8]	G. Keller and C. Liverani., Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141. Google Scholar
[9]	A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. Google Scholar
[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. Google Scholar
[11]	Z. Li, W-like maps with various instabilities of acim's,, Preprint, (). Google Scholar
[12]	T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177. Google Scholar
[13]	R. Murray, Ulam's method for some non-uniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007. doi: 10.3934/dcds.2010.26.1007. Google Scholar
[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/S0362-546X(99)00329-6. Google Scholar
[15]	S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960). Google Scholar

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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. Google Scholar
[2]	A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,", Probability and its Applications, (1997). doi: 10.1007/978-1-4612-2024-4. Google Scholar
[3]	Jiu Ding and Aihui Zhou, "Statistical Properties of Deterministic Systems,", Tsinghua University Texts, (2009). doi: 10.1007/978-3-540-85367-1. Google Scholar
[4]	P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for piecewise monotonic transformations,, Preprint, (). Google Scholar
[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. Google Scholar
[6]	P. Góra, On small stochastic perturbations of mappings of the unit interval,, Colloq. Math., 49 (1984), 73. Google Scholar
[7]	G. Keller, Stochastic stability in some chaotic dynamical systems,, Monatshefte für Mathematik, 94 (1982), 313. doi: 10.1007/BF01667385. Google Scholar
[8]	G. Keller and C. Liverani., Stability of the spectrum for transfer operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141. Google Scholar
[9]	A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. Google Scholar
[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. Google Scholar
[11]	Z. Li, W-like maps with various instabilities of acim's,, Preprint, (). Google Scholar
[12]	T. Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture,, Jour. Approx. Theory, 17 (1976), 177. Google Scholar
[13]	R. Murray, Ulam's method for some non-uniformly expanding maps,, Discrete and Continuous Dynamical Systems, 26 (2010), 1007. doi: 10.3934/dcds.2010.26.1007. Google Scholar
[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/S0362-546X(99)00329-6. Google Scholar
[15]	S. M. Ulam, "A Collection of Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, (1960). Google Scholar

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