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Summability implies Collet–Eckmann almost surely

Published online by Cambridge University Press:  21 January 2013

BING GAO  and
WEIXIAO SHEN
BING GAO
Affiliation:
Block S17, 10 Lower Kent Ridge Road, Singapore 119076, [email protected]@nus.edu.sg
WEIXIAO SHEN
Affiliation:
Block S17, 10 Lower Kent Ridge Road, Singapore 119076, [email protected]@nus.edu.sg
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Abstract

We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family ${f}_{t} $ of maps with ${f}_{0} = f$, we prove that $t= 0$ is a Lebesgue density point of the set of parameters for which ${f}_{t} $ satisfies both the Collet–Eckmann condition and a strong polynomial recurrence condition.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1184 - 1209
Copyright
Copyright © Cambridge University Press 

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References

Avila, A.. Infinitesimal perturbations of rational maps. Nonlinearity 15 (2002), 695–704.Google Scholar
Avila, A. and Moreira, C. G.. Statistical properties of unimodal maps: smooth families with negative Schwarzian devirative. Astérisque 268 (2003), 81–118.Google Scholar
Avila, A. and Moreira, C. G.. Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2) 161 (2005), 831–881.Google Scholar
Benedicks, M. and Carleson, L.. On iterations of $1- a{x}^{2} $ on $(- 1, 1)$. Ann. of Math. (2) 122 (1985), 1–24.Google Scholar
Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. of Math. (2) 133 (1991), 73–169.Google Scholar
Collet, P. and Eckmann, J. P.. Positive Lyapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13–46.Google Scholar
Jakobson, M.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981), 39–88.Google Scholar
Levin, G.. On an analytic approach to the Fatou conjecture. Fund. Math. 171 (2002), 177–196.CrossRefGoogle Scholar
Levin, G.. Perturbations of weakly expanding postcritical orbits. Preprint, 2011.Google Scholar
Levin, G.. Multipliers of periodic orbits in spaces of rational maps. Ergod. Th. & Dynam. Sys. 31 (2011), 197–243.Google Scholar
Luzzatto, S.. Bounded recurrence of critical points and Jakobson's theorem. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Cambridge University Press, Cambridge, 2000, pp. 173–210.Google Scholar
Lyubich, M.. Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures. Astérisque 261 (2000), 173–200.Google Scholar
Lyubich, M.. Almost every real quadratic map is either regular or stochastic. Ann. of Math. (2) 156 (2002), 1–78.Google Scholar
Nowicki, T. and van Strien, S.. Invariant measures exist under a summability condition for unimodal maps. Invent. Math. 105 (1991), 123–136.Google Scholar
Rees, M.. Positive measure sets of ergodic rational maps. Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 383–407.CrossRefGoogle Scholar
Shen, W.. On stochastic stability of non-uniformly expanding interval maps. Prepint, 2011, ArXiv:1107.2537.Google Scholar
Thieullen, P., Tresser, C. and Young, L.-S.. Positive Lyapunov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64 (1994), 121–172.Google Scholar
Tsujii, M.. Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math. 111 (1993), 113–137.CrossRefGoogle Scholar
Viana, M.. Multidimensional non-hyperbolic attractors. Publ. Math. Inst. Hautes Études Sci. 85 (1997), 63–96.Google Scholar
Wang, Q. and Takahasi, H.. Non-uniformly expanding 1D maps with logarithmic singularities. Nonlinearity 25 (2012), 533–550.Google Scholar
Yoccoz, J.-C.. Dynamique des polynômes quadratiques (Notes prepared by M. Flexor) (Panor. Synthéses, 8, Dynamique et géométrie complexes, Lyon, 1997). Société Mathématique de France, Paris, 1999, pp. x, xii, 187–222.Google Scholar