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Decay of correlations in one-dimensional dynamics. (English. French summary) Zbl 1039.37021
Summary: We consider multimodal \(C^3\) interval maps \(f\) satisfying a summability condition on the derivatives \(D_n\) along the critical orbits which implies the existence of an absolutely continuous \(f\)-invariant probability measure \(\mu\). If \(f\) is non-renormalizable, \(\mu\) is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence \((D_n)\) as \(n\to \infty\). We also give sufficient conditions for \(\mu\) to satisfy the central limit theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
60F05 Central limit and other weak theorems
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References:
[1] Baladi V. , Viana M. , Strong stochastic stability and rate of mixing for unimodal maps , Ann. Sci. Éc. Norm. Sup. 29 ( 1996 ) 483 - 517 . Numdam | MR 1386223 | Zbl 0868.58051 · Zbl 0868.58051 · numdam:ASENS_1996_4_29_4_483_0 · eudml:82414
[2] Benedicks M. , Carleson L. , On iterations of x \rightarrowtail 1 - ax 2 on ( - 1,1) , Ann. Math. 122 ( 1985 ) 1 - 25 . Zbl 0597.58016 · Zbl 0597.58016 · doi:10.2307/1971367
[3] Benedicks M. , Carleson L. , The dynamics of the Hénon map , Ann. Math. 133 ( 1991 ) 73 - 169 . MR 1087346 | Zbl 0724.58042 · Zbl 0724.58042 · doi:10.2307/2944326
[4] Blokh A. , Lyubich M. , Measurable dynamics of S-unimodal maps , Ann. Sci. Éc. Norm. Sup. 24 ( 1991 ) 545 - 573 . Numdam | MR 1132757 | Zbl 0790.58024 · Zbl 0790.58024 · numdam:ASENS_1991_4_24_5_545_0 · eudml:82305
[5] Bruin H. , Luzzatto S. , van Strien S. , Decay of correlation in one-dimensional dynamics , Preprint IHÉS, 1999. arXiv | MR 1858488 | Zbl 1039.37021 · Zbl 1039.37021 · doi:10.1016/S0012-9593(03)00025-9 · numdam:ASENS_2003_4_36_4_621_0 · eudml:82613
[6] Bruin H. , Keller G. , Equilibrium states for S-unimodal maps , Ergodic Theory Dynam. Systems 18 ( 1998 ) 765 - 789 . MR 1645373 | Zbl 0916.58020 · Zbl 0916.58020 · doi:10.1017/S0143385798108337
[7] Bruin H. , van Strien S. , Expansion of derivatives in one-dimensional dynamics , Israel J. Math. (to appear). MR 2013358 | Zbl 1122.37310 · Zbl 1122.37310 · doi:10.1007/BF02785964
[8] Bruin H. , van Strien S. , Existence of acips for multimodal maps , in: Global Analysis of Dynamical Systems, Festschrift to Floris Takens for his 60th birthday , IOP Publishing , Bristol , 2001 , pp. 433 - 447 . Zbl 01702976 · Zbl 1198.37066
[9] Collet P. , Statistics of closes return times for some non uniformly hyperbolic systems , Ergodic Theory Dynam. Systems 21 ( 2001 ) 401 - 420 . MR 1827111 | Zbl 1002.37019 · Zbl 1002.37019 · doi:10.1017/S0143385701001201
[10] Guckenheimer J. , Sensitive dependence on initial conditions for unimodal maps , Comm. Math. Phys. 70 ( 1979 ) 133 - 160 . Article | MR 553966 | Zbl 0429.58012 · Zbl 0429.58012 · doi:10.1007/BF01982351 · minidml.mathdoc.fr
[11] Jakobson M.V. , Absolutely continuous invariant measures for one-parameter families of one-dimensional maps , Comm. Math. Phys. 81 ( 1981 ) 39 - 88 . Article | MR 630331 | Zbl 0497.58017 · Zbl 0497.58017 · doi:10.1007/BF01941800 · minidml.mathdoc.fr
[12] Keller G. , Exponents, attractors, and Hopf decompositions for interval maps , Ergodic Theory Dynam. Systems 10 ( 1990 ) 717 - 744 . MR 1091423 | Zbl 0715.58020 · Zbl 0715.58020 · doi:10.1017/S0143385700005861
[13] Keller G. , Nowicki T. , Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps , Comm. Math. Phys. 149 ( 1992 ) 31 - 69 . Article | MR 1182410 | Zbl 0763.58024 · Zbl 0763.58024 · doi:10.1007/BF02096623 · minidml.mathdoc.fr
[14] Kozlovski O. , Getting rid of the negative Schwarzian derivative condition , Ann. Math. 152 ( 2000 ) 743 - 762 . MR 1815700 | Zbl 0988.37044 · Zbl 0988.37044 · doi:10.2307/2661353 · www.math.princeton.edu · eudml:121874
[15] Ledrappier F. , Some properties of absolutely continuous measures of an interval , Ergodic Theory Dynam. Systems 1 ( 1981 ) 77 - 93 . MR 627788 | Zbl 0487.28015 · Zbl 0487.28015 · doi:10.1017/S0143385700001176
[16] Liverani C. , Saussol B. , Vaienti S. , A probabilistic approach to intermittency , Ergodic Theory Dynam. Systems 19 ( 1999 ) 671 - 686 . MR 1695915 | Zbl 0988.37035 · Zbl 0988.37035 · doi:10.1017/S0143385799133856
[17] Lyubich M. , Milnor J. , The Fibonacci unimodal map , J. Amer. Math. Soc. 6 ( 1993 ) 425 - 457 . MR 1182670 | Zbl 0778.58040 · Zbl 0778.58040 · doi:10.2307/2152804
[18] Mañé R. , Hyperbolicity, sinks and measure in one dimensional dynamics , Comm. Math. Phys. 100 ( 1985 ) 495 - 524 . Article | MR 806250 | Zbl 0583.58016 · Zbl 0583.58016 · doi:10.1007/BF01217727 · minidml.mathdoc.fr
[19] de Melo W. , van Strien S. , One-Dimensional Dynamics , Springer , 1993 . MR 1239171 | Zbl 0791.58003 · Zbl 0791.58003
[20] Misiurewicz M. , Absolutely continuous measures for certain maps of an interval , Publ. IHÉS 53 ( 1981 ) 17 - 51 . Numdam | MR 623533 | Zbl 0477.58020 · Zbl 0477.58020 · doi:10.1007/BF02698686 · numdam:PMIHES_1981__53__17_0 · eudml:103973
[21] Nowicki T. , Sands D. , Non-uniform hyperbolicity and universal bounds for S -unimodal maps , Invent. Math. 132 ( 1998 ) 633 - 680 . MR 1625708 | Zbl 0908.58016 · Zbl 0908.58016 · doi:10.1007/s002220050236
[22] Nowicki T. , van Strien S. , Absolutely continuous measures under a summability condition , Invent. Math. 105 ( 1991 ) 123 - 136 . MR 1109621 | Zbl 0736.58030 · Zbl 0736.58030 · doi:10.1007/BF01232258 · eudml:143905
[23] Przytycki F. , Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures , Trans. Amer. Math. Soc. 350 ( 1998 ) 717 - 742 . MR 1407501 | Zbl 0892.58063 · Zbl 0892.58063 · doi:10.1090/S0002-9947-98-01890-X
[24] van Strien S. , Vargas E. , Real bounds, ergodicity and negative Schwarzian for multimodal maps , Preprint, 2000 and 2001. · Zbl 1073.37043
[25] Thunberg H. , Positive Lyapunov exponents for maps with flat critical points , Ergodic Theory Dynam. Systems ( 1998 ) 767 - 807 . MR 1695920 | Zbl 0966.37011 · Zbl 0966.37011 · doi:10.1017/S0143385799130177
[26] Tsujii M. , Small random perturbations of one-dimensional dynamical systems and Margulis-Pesin entropy formula , Random Comput. Dynam. 1 ( 1992 ) 59 - 89 . MR 1181380 | Zbl 0783.58018 · Zbl 0783.58018
[27] Tsujii M. , Positive Lyapunov exponents in families of one-dimensional dynamical systems , Invent. Math. 111 ( 1993 ) 113 - 137 . MR 1193600 | Zbl 0787.58029 · Zbl 0787.58029 · doi:10.1007/BF01231282 · eudml:144072
[28] Vargas E. , Measure of minimal sets of polymodal maps , Ergodic Theory Dynam. Systems 16 ( 1996 ) 159 - 178 . MR 1375131 | Zbl 0851.58015 · Zbl 0851.58015 · doi:10.1017/S0143385700008750
[29] Viana M. , Stochastic Dynamics of Deterministic Systems , Lecture Notes , 21 , Braz. Math. Colloqium , 1997 .
[30] Young L.-S. , Decay of correlations of certain quadratic maps , Comm. Math. Phys. 146 ( 1992 ) 123 - 138 . Article | MR 1163671 | Zbl 0760.58030 · Zbl 0760.58030 · doi:10.1007/BF02099211 · minidml.mathdoc.fr
[31] Young L.-S. , Statistical properties of dynamical systems with some hyperbolicity , Ann. of Math. 147 ( 1998 ) 585 - 650 . MR 1637655 | Zbl 0945.37009 · Zbl 0945.37009 · doi:10.2307/120960
[32] Young L.-S. , Recurrence times and rates of mixing , Israel J. Math. 110 ( 1999 ) 153 - 188 . MR 1750438 | Zbl 0983.37005 · Zbl 0983.37005 · doi:10.1007/BF02808180
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