Abstract
We prove exponential decay of correlations for (f, μ), wheref belongs in a positive measure set of quadratic maps of the interval and μ is its absolutely continuous invariant measure. These results generalize to other interval maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BC1] Benedicks, M., Carleson, L.: On iterations of 1−ax 2 on (−1,1). Ann. Math.122, 1–25 (1985)
[BC2] Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math.133, 73–169 (1991)
[BY] Benedicks, M., Young, L.-S.: Absolutely continuous invariant measures and random perturbations of certain one-dimensional maps to appear in Ergod. Theoret. Dynam. Sys.
[BS] Bunimovich, L.A., Sinai, Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys.78, 479–497 (1981)
[C] Collet, P.: Ergodic properties some unimodal mappings of the interval, 1984 preprint
[CE] Collet, P., Eckmann, J.-P.: On the abundance of aperiodic behavior for maps on the interval. Commun. Math. Phys.73, 115–160 (1980)
[DS] Dunford, Schwartz: Linear operators, Part I: General Theory 1957
[HK] Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math Z. 119–140 (1982)
[J] Jakobson, M.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys.81, 39–88 (1981)
[K] Keller, G.: Un théoreme de la limite centrale pour une classe de transformations monotones par morceaux. C. R. Acad. Sci. Paris, t.291 Série A, 155–158 (1980)
[LY] Lasota, A., Yorke, J.: On the existence of invariant measures for piecewise monotonic transformations. Trans. AMS186, 481–488 (1973)
[L] Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. Dynam. Sys.1, 77–93 (1981)
[M] Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES53, 17–51 (1981)
[N] Nowicki, T.: Symmetric S-unimodal mappings and positive Lyapunov exponents. Ergod. Th. Dynam. Sys.5, 611–616 (1985)
[Ro] Rohlin, V.A.: Lectures on the theory of entropy of transformations with invariant measures. Russ. Math. Surv.22(5), 1–54 (1967)
[Ru1] Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison-Wesley 1978
[Ru2] Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys.125, 239–262 (1989)
[Ry1] Rychlik, M.: Bounded variation and invariant measures. Studia Math.LXXVI, 69–80 (1983)
[Ry2] Rychlik, M.: Another proof of Jacobson's theorem and related results. Ergod. Th Dynam. Sys.9, 93–109 (1988)
[TTY] Thieullen, P., Tresser, C., Young, L.-S.: Pisitive exponent for generic 1-parameter families of 1-d maps, 1991 preprint
[Z] Ziemian, K.: On Perron-Frobenius operator and limit theorems for some maps of an interval. Ergod. Th. Rel. Topics II, Proceedings, Teubner Texte zür Math.94, 206–211 (1987)
Author information
Authors and Affiliations
Additional information
Communicated by T. Spencer
The results in this paper are announced in the Tagungsbericht of Oberwolfach, June 1990
The author is partially supported by NSF
Rights and permissions
About this article
Cite this article
Young, L.S. Decay of correlations for certain quadratic maps. Commun.Math. Phys. 146, 123–138 (1992). https://doi.org/10.1007/BF02099211
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02099211