Positive Lyapunov exponents in families of one dimensional dynamical systems

Tsujii, Masato

doi:10.1007/bf01231282

Cited by 72 publications

(93 citation statements)

References 11 publications

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“…Corollary 5.8, in the case d = 2, was obtained in [J]. The generalization to the higher degree case is well known (see [T,Theorem 2], which follows the approach of [BC]). Our proof is rather different.…”

Section: Real Parametersmentioning

confidence: 82%

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Ávila¹,

Lyubich²,

Shen³

2010

J. Eur. Math. Soc.

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We study the parameter space of unicritical polynomials fc : z → z d +c. For complex parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c, the map fc is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the "principal nest" of parapuzzle pieces.

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Section: Real Parametersmentioning

confidence: 82%

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Ávila¹,

Lyubich²,

Shen³

2010

J. Eur. Math. Soc.

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“…A similar condition played an important role in the work of Tsujii [41] in generalizing parameter exclusion arguments to neighbourhood of quite general maps. Again however we do feel that this is more of a technical simplifying assumption rather than a deep condition.…”

Section: 1mentioning

confidence: 99%

“…The fact that it has positive measure is therefore non-trivial. This was first shown in the ground-breaking work of Jakobson [16] and was generalized, over the years, in several papers; we mention in particular [6,10,13,14,33,36,39,41,45] for smooth maps with non-degenerate critical points, [40] for maps with a degenerate (flat) critical point, and [23,24] for maps with both critical points and singularities with unbounded derivatives.…”

Section: Introductionmentioning

confidence: 99%

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Luzzatto

Takahasi

2006

Nonlinearity

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Abstract. We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurrence of nonuniform hyperbolicity (stochastic dynamics) in families of onedimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family f a (x) = x 2 − a which have an absolutely continuous invariant probability measure is at least 10 −5000 .

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“…A first and remarkable observation which was already made as far back as the 1940's by Ulam and von Neumann [30] is that the parameter 2 can be thought of as "stochastic" in a precise mathematical sense (more formally, the corresponding map f a admits an ergodic invariant probability measure which is absolutely continuous with respect to Lebesgue measure). In 1981, Jakobson [6] and then in 1985 Benedicks and Carleson [2], showed that the set Ω + ⊂ Ω of such stochastic parameters is actually "large" in the sense that it has positive Lebesgue measure (see also some generalizations in [24,28,23,20,27,12,13]). Interestingly, however, it is also "small" in the sense that it is topologically nowhere dense, a fact that follows from the remarkable result that the set Ω − ⊂ Ω of "regular" parameters (for which almost every initial condition is eventually attracted to a periodic orbit) is open and dense in Ω [4,15,16] (see also generalizations in [10,11]).…”

Section: Background and Basic Definitionsmentioning

confidence: 99%

Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family

Golmakani

Luzzatto

Pilarczyk

2015

Experimental Mathematics

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Abstract. We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We compute a possibly small radius of the critical neighborhood, and a lower bound for the corresponding expansivity exponent outside this neighborhood, valid for all the parameters in each of the intervals. We illustrate and study the distribution of the radii and these exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.Key words and phrases: dynamical system, quadratic map family, uniform expansivity.We report on some rigorous numerical results concerning expansivity exponents for various parameter intervals for the dynamics outside a critical neighborhood in the quadratic family of maps. For completeness and in order to motivate our study, we start in Section 1 with some general facts on the quadratic family. The reader who is already familiar with the basic background may wish to skip directly to Section 2.

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Positive Lyapunov exponents in families of one dimensional dynamical systems

Cited by 72 publications

References 11 publications

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family

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