Blank M, Keller G (1997)
Publication Type: Journal article
Publication year: 1997
Publisher: Institute of Physics: Hybrid Open Access
Book Volume: 10
Pages Range: 81--107
Journal Issue: 1
DOI: 10.1088/0951-7715/10/1/006
We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the so-called bounded variation approach combined with the ergodic theorem of Ionescu - Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of `traps' under the action of random perturbations.
APA:
Blank, M., & Keller, G. (1997). Stochastic stability versus localization in one-dimensional chaotic dynamical systems. Nonlinearity, 10(1), 81--107. https://dx.doi.org/10.1088/0951-7715/10/1/006
MLA:
Blank, Michael, and Gerhard Keller. "Stochastic stability versus localization in one-dimensional chaotic dynamical systems." Nonlinearity 10.1 (1997): 81--107.
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