Stochastic perturbations and Ulam'smethod for W-shaped maps

Paweł Góra; Abraham Boyarsky

doi:10.3934/dcds.2013.33.1937

Article Contents

2013, Volume 33, Issue 5: 1937-1944. Doi: 10.3934/dcds.2013.33.1937

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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

Received: November 2011

Revised: January 2012

Published: May 2013

Abstract / Introduction Related Papers Cited by

Abstract

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:

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.

Mathematics Subject Classification: Primary: 37A10; Secondary: 37A05, 37E05.

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References

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