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

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.
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.

[1]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007

[2]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35

[3]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101

[4]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877

[5]

Xavier Bressaud. Expanding interval maps with intermittent behaviour, physical measures and time scales. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 517-546. doi: 10.3934/dcds.2004.11.517

[6]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451

[7]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100

[8]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013

[9]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

[10]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[11]

Magnus Aspenberg, Viviane Baladi, Juho Leppänen, Tomas Persson. On the fractional susceptibility function of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 679-706. doi: 10.3934/dcds.2021133

[12]

James P. Kelly, Kevin McGoff. Entropy conjugacy for Markov multi-maps of the interval. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2071-2094. doi: 10.3934/dcds.2020353

[13]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105

[14]

Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753

[15]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885

[16]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163

[17]

Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365

[18]

Peyman Eslami. Inducing schemes for multi-dimensional piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 353-368. doi: 10.3934/dcds.2021120

[19]

Carlangelo Liverani. A footnote on expanding maps. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741

[20]

Oliver Butterley. An alternative approach to generalised BV and the application to expanding interval maps. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3355-3363. doi: 10.3934/dcds.2013.33.3355

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]