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Tsujii, Masato

Positive Lyapunov exponents in families of one dimensional dynamical systems. (English) Zbl 0787.58029

Invent. Math. 111, No. 1, 113-137 (1993).
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)

Cited in 2 Reviews
Cited in 26 Documents

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37E05 Dynamical systems involving maps of the interval
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
28D05 Measure-preserving transformations

Keywords:

one-parameter family; absolutely continuous invariant measures; one-dimensional dynamical systems; density

Citations:

Zbl 0497.58017
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\textit{M. Tsujii}, Invent. Math. 111, No. 1, 113--137 (1993; Zbl 0787.58029)
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References:

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[2] [BC2] Benedicks, M., Carleson, L.: The Dynamics of the Hénon Map. Ann. Math., II. Ser.133, 73-169 (1991) · Zbl 0724.58042
[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)
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[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
[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
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