Abstract
Given a non-flat S-unimodal interval map f, we show that there exists C which only depends on the order of the critical point c such that if |Df n(f(c))|≥C for all n sufficiently large, then f admits an absolutely continuous invariant probability measure (acip). As part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small, then f admits an acip. As a special case, any S-unimodal map with critical order ℓ<2+ɛ having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of Nowicki & van Strien [21] and Martens & Nowicki [17] can be weakened considerably.
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References
Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1–17 (1979)
Bruin, H.: Topological conditions for the existence of invariant measures for unimodal maps. Ergod. Th. & Dynam. Sys. 14, 433–451 (1994)
Bruin, H.: Topological conditions for the existence of Cantor attractors. Trans. Am. Math. Soc. 350, 2229–2263 (1998)
Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996)
Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dyn. Sys. 3, 13–46 (1983)
Graczyk, J., Świtek, G.: Induced expansion for quadratic polynomials. Ann. Sci Éc. Norm. Súp. 29, 399–482 (1996)
Graczyk, J., Sands, D., Świtek, G.: Decay of geometry for unimodal maps: Negative Schwarzian case. Preprint, 2000
Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)
Jakobson, M., Świtek, G.: Metric properties of non-renormalizable S-unimodal maps. I. Induced expansion and invariant measures. Ergod. Th. & Dynam. Sys. 14, 721–755 (1994)
Johnson, S.: Singular measures without restrictive intervals. Commun. Math. Phys. 110, 185–190 (1987)
Keller, G., Nowicki, T.: Fibonacci maps re(aℓ)visited. Ergod. Th. & Dyn. Sys. 15, 99–120 (1995)
Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140, 347–404 (1994) and Erratum Manuscript, 2000
Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6, 425–457 (1993)
Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100(4), 495–524 (1985)
Martens, M.: Interval dynamics. Ph.D. Thesis, Delft, 1990
Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14, 331–349 (1994)
Martens, M., Nowicki, T.: Invariant measures for typical quadratic maps, Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque 261, 239–252 (2000)
de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin-Heidelberg-New York: Springer, 1993
Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. I.H.E.S. 53, 17–51 (1981)
Nowicki, T.: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergod. Th. & Dynam. Sys. 8, 425–435 (1988)
Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition. Invent. Math. 105, 123–136 (1991)
Pianigiani, G.: Absolutely continuous invariant measures on the interval for the process x n+1=Ax n (1-x n ). Boll. Un. Mat. Ital. 16, 364–378 (1979)
Shen, W.: Decay geometry for unimodal maps: An elementary proof. Preprint Warwick, 2002
Straube, E.: On the existence of invariant absolutely continuous measures. Commun. Math. Phys. 81, 27–30 (1981)
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Communicated by P. Sarnak
HB was supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW)
WS was supported by EPSRC grant GR/R73171/01
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Bruin, H., Shen, W. & Strien, S. Invariant Measures Exist Without a Growth Condition. Commun. Math. Phys. 241, 287–306 (2003). https://doi.org/10.1007/s00220-003-0928-z
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DOI: https://doi.org/10.1007/s00220-003-0928-z