The Pseudo-Orbit Shadowing Property for Markov Operators in the Space of Probability Density Functions

Boyarsky, Abraham ; Góra, Pawel

Canadian journal of mathematics, Tome 42 (1990) no. 6, pp. 1000-1017

Résumé

Let X be a space with two metrics d1 and d2. Let S :(X, d1 ) → (X, d 2) be continuous. We say S has the generalized pseudoorbit shadowing property with respect to the metrics d 1 and d 2 if for every every δ-pseudo-orbit in d1 can be ∊-shadowed by a true orbit in d 2, i.e., if {x 0, x 1,...} satisfies for all i ≧ 0, then for all i ≧ 0. The main result of this note shows that certain Markov operators P : L1→ L1 have the generalized shadowing property on weakly compact subsets of the space of probability density functions, where d 1 is the metric of norm convergence and d 2 is the metric of weak convergence. An important class of such operators are the Frobenius-Perron operators induced by certain expanding and nonexpanding maps on the interval. When there is exponential convergence of the iterates to the density, we can express δ in terms of ∊. We also show that, unlike the situation in the space X itself, the generalized shadowing property is valid for all parameters in families of maps and that there is stability of the shadowing property.

Détail

DOI : 10.4153/CJM-1990-053-2

Mots-clés : 58F11, 28D05

Bibliographie

[1] 1. Bowen, R., On axiom A diffeomorphisms, CBMS Regional Conference Series in Math., No. 35, 1978. Google Scholar

[2] 2. Coven, E.M., Kan, I., and Yorke, J.A., Pseudo-orbit shadowing in the family of tent maps, preprint 1986. Google Scholar

[3] 3. Lasota, A. and Mackey, M.C., Probabilistic properties and deterministic systems, ambridge University Press, 1985.10.1017/CBO9780511897474 Google Scholar | DOI

[4] 4. Bauer, W. and Sigmund, K., Topological dynamics of transformations induced in the space of probability measures, Monatsh fur Math., 79, 81-92, 1975. Google Scholar

[5] 5. Komuro, M., The pseudo orbit tracing properties in the space of probability measures, Tokyo J. Math., Vol. 72, No. 2, 461-468, 1984. Google Scholar

[6] 6. Milnor, J., On the concept of attractor: correction and remarks. Comm. Math. Phys. 102, 517- 519, 1985. Google Scholar

[7] 7. Lasota, A. and Yorke, J.A., On the existence of invariant measures for piecewise monotonie transformations, Trans. Amer. Math. Soc., V.116, 481-488, 1973. Google Scholar

[8] 8. Komornik, J., Asymptotic periodicity of the interates of weakly constrictive Markov operators , Tohoku Math. Journ. 38, 15-27, 1986. Google Scholar

[9] 9. Keller, G., Stochastic stability in some chaotic dynamical systems, Monatsh. fur Math. 94, 313- 333, 1983. Google Scholar

[10] 10. Keller, G. (personal communication). Google Scholar

[11] 11. Hofbauer, F. and Keller, G., Ergodic properties of invariant measures for piecewise monotonie transformations, Math. Z. 180, 119-140, 1982. Google Scholar

[12] 12. Ionescu Tulcea, C. T., and Marinescu, G., Theotie ergodique pour les classes d'opérations non complètement continues. Ann. Math. 52, 140-147, 1950. Google Scholar

[13] 13. Lasota, A., Li, T.Y. and Yorke, J.A., Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc, 286, 751-764, 1984. Google Scholar

[14] 14. Pelikan, S., Invariant densities for random maps of the interval, Trans. Amer. Math. Soc, 281, 813-825, 1984. Google Scholar

[15] 15. Boyarsky, A., Uniqueness of invariant densities for certain random maps of the interval, Can. Math. Bull., Vol. 30 (3), 301-308, 1987. Google Scholar

[16] 16. Neveu, J., Mathematical foundations of the calculus of probability, Holden-day, 1965. Google Scholar

[17] 17. Szewc, B., The Perron-Frobenius operator in spaces on smooth functions, Erg. Theory and Dyn. Syst., 4, No. 4, 613-643, 1984. Google Scholar

[18] 18. Keller, G., Generalized bounded variation and applications to piecewise monotonie transformations, Z.|Wahr. 69, 461-478, 1985. Google Scholar