Rational Misiurewicz Maps are Rare

Aspenberg, Magnus

doi:10.1007/s00220-009-0813-5

Cited by 12 publications

(36 citation statements)

References 18 publications

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“…Following Aspenberg's idea in [1] we will show that δ-Misiurewicz parameters are rare in any neighbourhood of λ 0 .…”

Section: Introductionmentioning

confidence: 82%

“…The proof of Theorem 1.3 in general follows the Aspenberg's approach from [1], however it differs in some crucial details. The main difficulty is the presence of essential singularity at ∞ and infinite degree of maps.…”

Section: Theorem 13mentioning

confidence: 99%

“…Now, we want to obtain so-called transversality condition (cf. [1]), which says that the asymptotic value 0 cannot follow its holomorphic motion h λ (0) in the whole parameter ball B(λ 0 , r). As we will see, it is an immediate consequence of the instability of Misiurewicz parameters (more general Collet-Eckmann maps) in the exponential family proved by Urbański and Zdunik in [15].…”

Section: Holomorphic Motion and Instabilitymentioning

confidence: 99%

“…the real quadratic family g a (x) = 1 − ax 2 in the case when g a is non-hyperbolic and the critical point 0 is non-recurrent. We refer to [1] for a nice discussion concerning various definitions of Misiurewicz condition in the complex case and more references. We are interested in defining Misiurewicz maps for the exponential family (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…One can ask about the Lebesgue measure of those parameters in the parameter space. Recently it was proved by M. Aspenberg in [1] that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions of any fixed degree. In this paper we prove the following.…”

Section: Introductionmentioning

confidence: 99%

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Misiurewicz parameters in the exponential family

Badeńska

2010

Math. Z.

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“…Following Aspenberg's idea in [1] we will show that δ-Misiurewicz parameters are rare in any neighbourhood of λ 0 .…”

Section: Introductionmentioning

confidence: 82%

Section: Theorem 13mentioning

confidence: 99%

Section: Holomorphic Motion and Instabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Misiurewicz parameters in the exponential family

Badeńska

2010

Math. Z.

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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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