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