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Absolutely continuous S.R.B. measures for random Lasota-Yorke maps

Author: Jérôme Buzzi
Journal: Trans. Amer. Math. Soc. 352 (2000), 3289-3303
MSC (2000): Primary 37A25, 37H99
Published electronically: March 24, 2000
MathSciNet review: 1707698
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Abstract: A. Lasota and J. A. Yorke proved that a piecewise expanding interval map admits finitely many ergodic absolutely continuous invariant probability measures. We generalize this to the random composition of such maps under conditions which are natural and less restrictive than those previously studied by Morita and Pelikan. For instance our conditions are satisfied in the case of arbitrary random $\beta $-transformations, i.e., $x\mapsto \beta x\mod 1$ on $[0,1]$ where $\beta $ is chosen according to any stationary stochastic process (in particular, not necessarily i.i.d.) with values in $]1,\infty [$.

RSESUM´E. A. Lasota et J. A. Yorke ont montré qu'une application de l'intervalle dilatante par morceaux admet un nombre fini de mesures de probabilité invariantes et ergodiques absolument continues. Nous généralisons ce résultat à la composition aléatoire de telles applications sous des conditions naturelles, moins restrictives que celles précédemment envisagées par Morita et Pelikan. Par exemple, nos conditions sont satisfaites par toute $\beta $-transformation aléatoire, i.e., $x\mapsto \beta x\mod 1$ sur $[0,1]$ avec $\beta $ choisi selon un processus stochastique stationnaire quelconque (en particulier, non-nécessairement i.i.d.) à valeurs dans $]1,\infty [$.

References [Enhancements On Off] (What's this?)

  • [1] V. Baladi, Correlation spectrum of quenched and annealed equilibrium states for random expanding maps, Commun. Math. Phys. 186 (1997), 671-700. MR 98i:58188
  • [2] V. Baladi, A. Kondah, B. Schmitt, Random correlations for small perturbations of expanding maps, Random Comput. Dynam. 4 (1996), 179-204. MR 97e:58139
  • [3] M. Blank, G. Keller, Stochastic stability versus localization in one-dimensional chaotic dynamical systems, Nonlinearity 10 (1997), 81-107. MR 98a:58101
  • [4] T. Bogenschütz, V.M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random Comput. Dynamics 1 (1992/93), 219-227. MR 93j:58042
  • [5] T. Bogenschütz, V.M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergod. Th. & Dynam. Sys. 15 (1995), 413-447. MR 96m:58133
  • [6] J. Buzzi, Absolutely continuous invariant measures for generic multi-dimensional piecewise expanding and affine maps, Int. J. Bifurcations & Chaos 9 (1999), 1743-1750.
  • [7] -, Exponential decay of correlations for random Lasota-Yorke maps, (preprint I.M.L. 98-12), to appear Commun. Math. Phys.
  • [8] -, A.c.i.m.'s for arbitrary expanding piecewise real-analytic mappings of the plane, (preprint I.M.L. 98-13), to appear Ergod. Th. & Dynam. Sys.
  • [9] -, A.c.i.m.'s as equilibrium states for piecewise invertible dynamical systems, (preprint I.M.L. 98).
  • [10] N. Dunford, T. Schwartz, Linear operators. Part I. General theory, Wiley, New York, 1958. MR 22:8302
  • [11] P. Ferrero, B. Schmitt, Produits aléatoires d'opérateurs matrices de transfert, Probab. Th. Related Fields 79 (1988), 227-248. MR 90e:47006
  • [12] P. Góra, A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformations in $R^{N}$, Israel J. Math. 67 (1989), 272-286. MR 91c:58061
  • [13] P. Góra, B. Schmitt, Un example de transformation dilatante et $C^{1}$ par morceaux de l'intervalle sans probabilité absolument continue invariante, Ergod. Th. & Dynam. Syst. 9 (1989), 101-113. MR 90h:58052
  • [14] K. Khanin, Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinai's Moscow Seminar on Dynamical Systems, A.M.S. Translation, Series 2, vol. 171, Amer. Math. Soc., Providence, Rhode Island, 1996, pp. 107-140. MR 46j:58136
  • [15] Yu. Kifer, Ergodic theory of random transformations, Birkhäuser, Boston, 1986. MR 89c:58069
  • [16] -, Random perturbations of dynamical systems, Birkhäuser, Boston, 1988. MR 91e:58159
  • [17] A. Kondah, Les endomorphismes dilatants de l'intervalle et leurs perturbations aléatoires, Thèse de l'Université de Bourgogne, Dijon, 1991.
  • [18] U. Krengel, Ergodic theorems, de Gruyter, Berlin, 1985. MR 87i:28001
  • [19] A. Lasota, J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. MR 49:538
  • [20] T. Morita, Random iteration of one-dimensional transformations, Osaka J. Math. 22 (1985), 489-518. MR 87a:58100
  • [21] S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc. 281 (1984), 813-825. MR 85i:58070
  • [22] M. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80. MR 85h:28019
  • [23] M. Tsujii, A.c.i.m. for piecewise real-analytic expanding mappings of the plane (preprint, Hokkaido, 1998).
  • [24] -, Piecewise $C^{r}$ expanding maps with singular ergodic properties (preprint, Hokkaido, 1998).

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Additional Information

Jérôme Buzzi
Affiliation: Institut de Mathématiques de Luminy, 163 av. de Luminy, Case 907, 13288 Marseille Cedex 9, France
Address at time of publication: Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Keywords: Absolutely continuous invariant measures; piecewise monotonic maps; random maps; transfer operator; bounded variation
Received by editor(s): January 1, 2025
Received by editor(s) in revised form: January 1, 1998
Published electronically: March 24, 2000
Additional Notes: Work partly done at the Laboratoire de Topologie de Dijon, Université de Bourgogne, France
Article copyright: © Copyright 2000 American Mathematical Society

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