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Invariant measures for interval maps with different one-sided critical orders

Published online by Cambridge University Press:  28 August 2013

HONGFEI CUI  and
YIMING DING
HONGFEI CUI
Affiliation:
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, PR China email [email protected]@wipm.ac.cn
YIMING DING
Affiliation:
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, PR China email [email protected]@wipm.ac.cn
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Abstract

For an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 835 - 853
Copyright
© Cambridge University Press, 2013 

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References

Araújo, V., Luzzatto, S. and Viana, M.. Invariant measures for interval maps with critical points and singularities. Adv. Math. 221 (5) (2009), 1428–1444.Google Scholar
Benedicks, M. and Misiurewicz, M.. Absolutely continuous invariant measures for maps with flat tops. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 203–213.CrossRefGoogle Scholar
Bruin, H., Rivera-Letelier, J., Shen, W. and van Strien, S.. Large derivatives, backward contraction and invariant densities for interval maps. Invent. Math. 172 (3) (2008), 509–533.Google Scholar
Bruin, H., Shen, W. and van Strien, S.. Invariant measures exist without a growth condition. Comm. Math. Phys. 241 (2–3) (2003), 287–306.Google Scholar
Collet, P. and Eckmann, J.-P.. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1) (1983), 13–46.Google Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25). Springer, Berlin, 1993.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T.. Linear Operators. Part I: General Theory. Wiley, New York, 1957.Google Scholar
Lasota, A. and Mackey, M. C.. Stochastic aspects of dynamics. Chaos, Fractals, and Noise (Applied Mathematical Sciences, 97), 2nd edn. Springer, New York, 1994.CrossRefGoogle Scholar
Luzzatto, S.. Stochastic-like behaviour in non-uniformly expanding maps. Handbook of Dynamical Systems. Vol. 1B. Eds. Hasselblat, B. and Katok, A.. Elsevier, 2006, pp. 265–326.Google Scholar
Mañé, R.. Hyperbolicity, sinks and measure in one-dimensional dynamics. Comm. Math. Phys. 100 (4) (1985), 495–524.CrossRefGoogle Scholar
Martens, M.. Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14 (2) (1994), 331–349.Google Scholar
Metzger, R. J.. Sinai–Ruelle–Bowen measures for contracting Lorenz maps and flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2) (2000), 247–276.Google Scholar
Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Publ. Math. Inst. Hautes Études Sci. (53) (1981), 17–51.Google Scholar
Nowicki, T. and van Strien, S.. Invariant measures exist under a summability condition for unimodal maps. Invent. Math. 105 (1) (1991), 123–136.CrossRefGoogle Scholar
Rivera-Letelier, J.. A connecting lemma for rational maps satisfying a no growth condition. Ergod. Th. & Dynam. Sys. 27 (2) (2007), 595–636.CrossRefGoogle Scholar
Rovella, A.. The dynamics of perturbations of the contracting Lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24 (2) (1993), 233–259.CrossRefGoogle Scholar
van Strien, S.. One-dimensional dynamics in the new millennium. Discrete Contin. Dyn. Syst. 27 (2) (2010), 557–588.Google Scholar