Misiurewicz parameters in the exponential family

Badeńska, Agnieszka

doi:10.1007/s00209-010-0671-z

Cited by 4 publications

(13 citation statements)

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“…equal to the dimension of the parameters space. For the exponential family f λ (z) = λe z , λ ∈ C similar results also hold, see [3] and [5]. We prove in this paper the following.…”

Section: Introductionsupporting

confidence: 70%

Semi-Hyperbolic Maps Are Rare

Aspenberg

2017

International Mathematics Research Notices

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“…equal to the dimension of the parameters space. For the exponential family f λ (z) = λe z , λ ∈ C similar results also hold, see [3] and [5]. We prove in this paper the following.…”

Section: Introductionsupporting

confidence: 70%

Semi-Hyperbolic Maps Are Rare

Aspenberg

2017

International Mathematics Research Notices

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“…The ideas in this section are not especially new, though the exposition and the formulation of results are. The reader may wish to compare this section with [3,Sections 3,4] and [1,Section 3]. The useful result is Lemma 37; it follows easily from the following proposition.…”

Section: Basic Parametric Estimatesmentioning

confidence: 98%

“…The estimates are also a little less cumbersome going forwards than backwards. For completeness, and because the desired estimates are not simple to extract from [3] (itself based on [1]), we include proofs of the estimates. Sectors of small annuli centred on λ 0 in parameter space get mapped biholomorphically and with bounded distortion onto reasonably large sets near P (f ) by the map λ → f n λ (0), for some n depending on the annulus, see Lemma 37.…”

Section: Structure Of the Proofmentioning

confidence: 99%

“…This result was superseded by many more in interval dynamics, see [7] for one of the latest and strongest. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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