×

zbMATH — the first resource for mathematics

Positive Lyapunov exponents in families of one dimensional dynamical systems. (English) Zbl 0787.58029
A generalization of Jakobson’s theorem [M. V. Jakobson, Commun. Math. Phys. 81, 39–88 (1981; Zbl 0497.58017)] concerning abundance of chaotic systems with absolutely continuous invariant measures in a typical family of one-dimensional dynamical systems is given. A new precise estimate for the density of the corresponding parameters is obtained.
Reviewer: H.D.Voulov (Sofia)

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [BC1] Benedicks, M., Carleson, L.: On Iterations of 1?ax 2 on (?1, 1). Ann. Math., II. Ser.122, 1-24 (1985) · Zbl 0597.58016 · doi:10.2307/1971367
[2] [BC2] Benedicks, M., Carleson, L.: The Dynamics of the Hénon Map. Ann. Math., II. Ser.133, 73-169 (1991) · Zbl 0724.58042 · doi:10.2307/2944326
[3] [BY] Benedicks, M., Young, L.-S.: Absolutely Continous Invariant Measures and Random Perturbations for Certain One Dimensional Maps. Ergodic Theory Dyn. Syst. (to appear)
[4] [BL] Blokh, A.M., Lyubich, M.Yu.: Non-existence of Wandering Interval and Structure of Topological Attractors of One Dimensional Dynamical Systems II. The Smooth Case. Ergodic Theory Dyn. Syst.,9, 751-758 (1989) · Zbl 0665.58024 · doi:10.1017/S0143385700005319
[5] [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 · doi:10.1007/BF01941800
[6] [Jo] Johnson, S.: Continuous Measures in One Dimension. Commun. Math. Phys.122, 293-320 (1989) · Zbl 0684.28008 · doi:10.1007/BF01257418
[7] [Le] Ledrappier, F.: Some Properties of Absolutely Continuous Invariant Measures on an Interval. Ergodic Theory Dyn. Sys.1, 77-93 (1981) · Zbl 0487.28015
[8] [Ma] Mañé, R.: Hyperbolicity, Sinks and Measure in One Dimensional Dynamics. Commun. Math. Phys.100, 495-524 (1985) · Zbl 0583.58016 · doi:10.1007/BF01217727
[9] [MMS] Martens, M., deMelo, W., van Strien, S.: Julia-Fatou-Sullivan Theory for One Dimensional Dynamics. Acta Math.168, 273-318 (1992) · Zbl 0761.58007 · doi:10.1007/BF02392981
[10] [MV] Mora, L., Viana, M.: Abundance of Strange Attractors. Acta. Math. (to appear) · Zbl 0815.58016
[11] [NS] Nowicki, T., van Strien, S.: Absolutely Continuous Measures for Collet-Eckmann Maps without Schwarzian Derivative Conditions. Invent. Math.93, 619-635 (1988) · Zbl 0659.58034 · doi:10.1007/BF01410202
[12] [R] Rychlik, M.: Another Proof of Jakobson’s Theorem and Related Results. Ergodic Theory Dyn. Syst.8, 93-110 (1986) · Zbl 0671.58019
[13] [T1] Tsujii, M.: Weak Regularity of Lyapunov Exponent in One Dimensional Dynamics. (Preprint)
[14] [T2] Tsujii, M.: A Proof of Benedicks-Carleson-Jakobson Theorem. (Preprint)
[15] [T3] Tsujii, M.: Small Random Perturbations of One Dimensional Dynamical Systems and Margulis-Pesin entropy Formula. Random Comput. Dyn.1, 59-89 (1992) · Zbl 0783.58018
[16] [Y] Young, L.-S.: Decay of Correlations for Certain Quadratic Maps. (Preprint) · Zbl 0760.58030
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.