Abstract
An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BC] Benedicks, M., Carleson, L.: On iterations of 1-ax 2 on (−1,1). Ann. Math.122, 1–25 (1983)
[B] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin, Heidelberg, New York: Springer 1975
[CE 1] Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Prog. in Phys., vol. 1. Boston, MA: Birkhäuser 1980
[CE 2] Collet, P., Eckman, J.-P.: Positive Ljapunov exponents and absolute continuity for maps on the interval. Ergod. Theor. Dyn. Sys.3, 13–46 (1981)
[DGS] Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, vol. 527. Berlin, Heidelberg, New York: Springer 1976
[F] Fischer, R.: Sofic systems and graphs. Monatshefte Math.80, 179–186 (1975)
[G] Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys.70, 133–160 (1979)
[H 1] Hofbauer, F.: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math.24, 213–237 (1979), Part II. Israel J. Math.38, 107–115 (1981)
[H 2] Hofbauer, F.: The topological entropy of the transformationx↦ax(1−x). Monatshefte Math.90, 117–141 (1980)
[H 3] Hofbauer, F.: Kneading invariants and Markov diagrams. In: Michel, H. (ed.). Ergodic theory and related topics. Proceedings, pp. 85–95. Berlin: Akademic-Verlag 1982
[Ja] Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys.81, 39–88 (1981)
[Jo] Johnson, S.: Singular measures without restrictive intervals. Commun. Math. Phys.110, 185–190 (1987)
[K 1] Keller, G.: Invariant measures and Lyapunov exponents forS-unimodal maps. Preprint Maryland 1987
[K 2] Keller, G.: Exponents, attractors, and Hopf decompositions for interval maps. To appear in Ergod. Theor. Dyn. Sys.
[K 3] Keller, G.: Lifting measures to Markov extensions. To appear in Monatshefte Math.
[L] Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Ergod. Theor. Dyn. Sys.1, 77–93 (1981)
[MT] Milnor, J., Thurston, W.: On iterated maps of the interval. Preprint Princeton 1977
[Mi] Misiurewicz, M.: Absolutely continuous invariant measures for certain maps of an interval. Publ. IHES53, 17–51 (1981)
[Ni] Nitecki, Z.: Topological dynamics on the interval. In: Ergodic theory and dynamical systems, vol. II. Katok, A. (ed.). Progress in Math, pp. 1–73, vol. 21. Boston: Birkhäuser 1982
[No] Nowicki, T.: SymmetricS-unimodal mappings and positive Liapunov exponents. Ergod. Theor. Dyn. Sys.5, 611–616 (1985)
[NvS] Nowicki, T., van Strien, S.: Absolutely continuous invariant measures forC 2-unimodal maps satisfying the Collet-Eckmann condition. Invent. Math.93, 619–635 (1988)
[P] Preston, C.: Iterates of maps on an interval. Lecture Notes in Mathematics, vol. 999. Berlin, Heidelberg, New York: Springer 1983
[R] Rychlik, M.: Another proof of Jakobson's theorem and related results. Ergod. Theor. Dynam. Sys.8, 93–109 (1988)
Author information
Authors and Affiliations
Additional information
Communicated by J.-P. Eckmann
Rights and permissions
About this article
Cite this article
Hofbauer, F., Keller, G. Quadratic maps without asymptotic measure. Commun.Math. Phys. 127, 319–337 (1990). https://doi.org/10.1007/BF02096761
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02096761