Positive exponent in families with flat critical point

Thunberg, Hans

doi:10.1017/s0143385799130177

Cited by 17 publications

(23 citation statements)

References 15 publications

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“…The concept of Misiurewicz parameter exists in other contexts too, see [14,6,3,10] for example. The articles [15,30,26,4] all find positive measure sets of non-hyperbolic parameters (indeed one admitting absolutely continuous invariant probability measures) in a neighbourhood of Misiurewicz parameters. On the other hand, Misiurewicz parameters have zero Lebesgue measure, in general ( [29,1,3]).…”

Section: Introductionmentioning

confidence: 99%

Perturbing Misiurewicz Parameters in the Exponential Family

Dobbs

2015

Commun. Math. Phys.

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In one-dimensional real and complex dynamics, a map whose postsingular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a non-hyperbolic map, as per Jakobson's Theorem for unimodal interval maps. This is despite genericity of hyperbolic parameters (at least in the interval setting). We show the contrary holds in the complex exponential family z → λ exp(z): Misiurewicz maps are Lebesgue density points for hyperbolic parameters. As a by-product, we also show that Lyapunov exponents almost never exist for exponential Misiurewicz maps. The lower Lyapunov exponent is −∞ almost everywhere. The upper Lyapunov exponent is non-negative and depends on the choice of metric.

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Section: Introductionmentioning

confidence: 99%

Perturbing Misiurewicz Parameters in the Exponential Family

Dobbs

2015

Commun. Math. Phys.

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“…Thunberg, in [33], showed Benedicks-Carleson type results for unimodal families of maps with critical behaviour like exp(−|x| −α ) for α < 1/8. He asked whether for α 1 no acip can exist.…”

Section: Smooth Maps With Flat Topsmentioning

confidence: 97%

On cusps and flat tops

Dobbs¹

2014

Ann. inst. Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to C 1+ǫ . The critical points are not required to verify a non-flatness condition, so the results are applicable to C 1+ǫ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

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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 exponent in families with flat critical point

Cited by 17 publications

References 15 publications

Perturbing Misiurewicz Parameters in the Exponential Family

Perturbing Misiurewicz Parameters in the Exponential Family

On cusps and flat tops

Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family

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