×

zbMATH — the first resource for mathematics

Quadratic maps without asymptotic measure. (English) Zbl 0702.58034
Summary: 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.

MSC:
37E05 Dynamical systems involving maps of the interval
37B99 Topological dynamics
MathOverflow Questions:
Physical measures that are not SRB
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [BC] Benedicks, M., Carleson, L.: On iterations of 1-ax 2 on (,1). Ann. Math.122, 1–25 (1983) · Zbl 0597.58016
[2] [B] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0308.28010
[3] [CE 1] Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Prog. in Phys., vol. 1. Boston, MA: Birkhäuser 1980 · Zbl 0458.58002
[4] [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)
[5] [DGS] Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, vol. 527. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0328.28008
[6] [F] Fischer, R.: Sofic systems and graphs. Monatshefte Math.80, 179–186 (1975) · Zbl 0314.54043
[7] [G] Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys.70, 133–160 (1979) · Zbl 0429.58012
[8] [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) · Zbl 0422.28015
[9] [H 2] Hofbauer, F.: The topological entropy of the transformationxx(1). Monatshefte Math.90, 117–141 (1980) · Zbl 0433.54009
[10] [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 · Zbl 0512.54022
[11] [Ja] Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys.81, 39–88 (1981) · Zbl 0497.58017
[12] [Jo] Johnson, S.: Singular measures without restrictive intervals. Commun. Math. Phys.110, 185–190 (1987) · Zbl 0641.58024
[13] [K 1] Keller, G.: Invariant measures and Lyapunov exponents forS-unimodal maps. Preprint Maryland 1987
[14] [K 2] Keller, G.: Exponents, attractors, and Hopf decompositions for interval maps. To appear in Ergod. Theor. Dyn. Sys. · Zbl 0715.58020
[15] [K 3] Keller, G.: Lifting measures to Markov extensions. To appear in Monatshefte Math. · Zbl 0712.28008
[16] [L] Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Ergod. Theor. Dyn. Sys.1, 77–93 (1981) · Zbl 0487.28015
[17] [MT] Milnor, J., Thurston, W.: On iterated maps of the interval. Preprint Princeton 1977
[18] [Mi] Misiurewicz, M.: Absolutely continuous invariant measures for certain maps of an interval. Publ. IHES53, 17–51 (1981) · Zbl 0477.58020
[19] [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 · Zbl 0506.54035
[20] [No] Nowicki, T.: SymmetricS-unimodal mappings and positive Liapunov exponents. Ergod. Theor. Dyn. Sys.5, 611–616 (1985) · Zbl 0615.28009
[21] [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) · Zbl 0659.58034
[22] [P] Preston, C.: Iterates of maps on an interval. Lecture Notes in Mathematics, vol. 999. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0582.58001
[23] [R] Rychlik, M.: Another proof of Jakobson’s theorem and related results. Ergod. Theor. Dynam. Sys.8, 93–109 (1988) · Zbl 0671.58019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.