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Harmonic average of slopes and the stability of absolutely continuous invariant measure

Title:

Harmonic average of slopes and the stability of absolutely continuous invariant measure

GÓRA, PAWEŁ, Li, Zhenyang and Boyarsky, Abraham (2012) Harmonic average of slopes and the stability of absolutely continuous invariant measure. Journal of Mathematical Analysis and Applications, 396 (1). pp. 1-6. ISSN 0022247X

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Official URL: http://dx.doi.org/10.1016/j.jmaa.2012.05.067

Abstract

For families of piecewise expanding maps which converge to a map with a fixed or periodic turning point touching a branch with slope of modulus equal to or less than 2, the standard Lasota–Yorke argument fails to prove stability. It is the goal of this paper to use instead the harmonic average of slopes condition for a large class of maps satisfying the summable oscillation condition for the reciprocal of the derivative. Using Rychlik’s Theorem for a family of perturbations we prove weak compactness in L1 of the density functions associated with them. From this it follows that we have stability of absolutely continuous invariant measures of the limit map.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:GÓRA, PAWEŁ and Li, Zhenyang and Boyarsky, Abraham
Journal or Publication:Journal of Mathematical Analysis and Applications
Date:2012
Digital Object Identifier (DOI):10.1016/j.jmaa.2012.05.067
Keywords:Absolutely continuous invariant measures; Stability of acim; Piecewise expanding maps of interval
ID Code:976822
Deposited By: DANIELLE DENNIE
Deposited On:29 Jan 2013 13:47
Last Modified:18 Jan 2018 17:43

References:

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[2] A. Lasota, J.A. Yorke On the existence of invariant measures for piecewise monotonic transformations Trans. Amer. Math. Soc., 186 (1973), pp. 481–488 MR0335758 (49 #538) (1974)

[3] A. Boyarsky, P. Góra Laws of Chaos, Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications, Birkhaüser, Boston, MA (1997)

[4] G. Keller, C. Liverani Stability of the spectrum for transfer operators Ann. Sc. Norm. Super Pisa Cl. Sci. (4), 28 (1) (1999), pp. 141–152

[5] P. Eslami, M. Misiurewicz Singular limits of absolutely continuous invariant measures for families of transitive map J. Difference Equ. Appl., 18 (4) (2012), pp. 739–750 http://dx.doi.org/10.1080/10236198.2011.590480

[6] Zhenyang Li, P. Góra, A. Boyarsky, H. Proppe, P. Eslami, A family of piecewise expanding maps having singular measure as a limit of ACIM’s, Ergodic Theory Dynam. Systems. http://dx.doi.org/10.1017/S0143385711000836 (in press).

[7] Zhenyang Li, W-like maps with various instabilities of acim’s. Available at http://arxiv.org/abs/1109.5199.

[8] P. Eslami, P. Góra, Stronger Lasota–Yorke inequality for piecewise monotonic transformations. Preprint.

[9] P. Collet, J.P. Eckmann Measures invariant under mappings of the unit interval ,in: G. Velo, A.S. Wightman (Eds.), Regular and Chaotic Motions in Dynamic Systems, NATO ASI Series, Series B: Physics, vol. 118, Plenum Press, NY (1985), pp. 233–265

[10] B. Schmitt, Contributions a l’étude de systemes dynamiques unidimensionnels en théorie ergodique, Ph.D. Thesis, University of Bourgogne, 1986.

[11] P. Góra Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative Ergodic Theory Dynam. Systems, 14 (3) (1994), pp. 475–492

[12] M.R. Rychlik, Invariant measures and the variational principle for Lozi mappings, Ph.D. Thesis, University of California, Berkeley, 1983.

[13] P. Góra, A. Boyarsky Compactness of invariant densities for families of expanding, piecewise monotonic transformations Canad. J. Math., XLI (1989), pp. 855–869
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