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

Title:

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

GÓRA, PAWEŁ and Boyarsky, Abraham (2012) Stochastic perturbations and Ulam's method for W-shaped maps. Discrete and Continuous Dynamical Systems, 33 (5). pp. 1937-1944. ISSN 1078-0947

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Official URL: http://dx.doi.org/10.3934/dcds.2013.33.1937

Abstract

For a discrete dynamical system given by a map τ:I→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 τ . 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 |τ ′ |>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 ± 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) μ a such that μ 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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:GÓRA, PAWEŁ and Boyarsky, Abraham
Journal or Publication:Discrete and Continuous Dynamical Systems
Date:2012
Digital Object Identifier (DOI):10.3934/dcds.2013.33.1937
Keywords:Piecewise expanding maps of an interval, absolutely continuous invariant measures, Frobenius-Perron operator, Markov maps, W-shaped maps, Ulam's method, harmonic average of slopes.
ID Code:976823
Deposited By: DANIELLE DENNIE
Deposited On:29 Jan 2013 13:52
Last Modified:18 Jan 2018 17:43

References:

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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.

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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.

14 R. Murray, Existence, mixing and approximation of invariant densities for expanding maps on , Nonlinear Analysis TMA, 45 (2001), 37-72.

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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