Hausdorff dimension of the hairs without endpoints for λ exp z

Karpińska, Bogusława

doi:10.1016/s0764-4442(99)80321-8

Cited by 54 publications

(50 citation statements)

References 5 publications

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“…Often, the Hausdorff dimension of all the rays is 1, while the endpoints alone have Hausdorff dimension 2, or even infinite planar Lebesgue measure. This "dimension paradox" was discovered by Karpińska for hyperbolic exponential maps [Kar99], for which the topology of Julia sets was known. In later extensions for arbitrary exponential maps [SZ03] and for the cosine family [RS08], the new parts were the topological classifications, while analogous results on the Hausdorff dimension followed from the methods of Karpińska and McMullen; see also [Sch07a], [Sch07b] for extreme results where every point in the complex plane is either on a dynamic ray (whose union still has dimension one) or a landing point of those raysso the landing points of this one-dimensional set of rays is the entire complex plane with only a one-dimensional set of exceptions.…”

Section: Corollary (Meromorphic Functions With Logarithmic Singularitmentioning

confidence: 97%

Dynamic rays of bounded-type entire functions

Rottenfußer¹,

et al. 2011

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We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative.On the other hand, we show that for many functions in B, in particular those of finite order, every escaping point can be connected to ∞ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

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Section: Corollary (Meromorphic Functions With Logarithmic Singularitmentioning

confidence: 97%

Dynamic rays of bounded-type entire functions

Rottenfußer¹,

et al. 2011

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“…The argument in [14] shows that under the hypothesis of Theorem F we also have dim(J(λ sin z)\E λ ) = 1.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 91%

Dynamics of a higher dimensional analog of the trigonometric functions

Bergweiler¹,

Erëmenko²

2011

Ann. Acad. Sci. Fenn. Math.

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Abstract. We introduce a quasiregular analog F of the sine and cosine function such that, for a sufficiently large constant λ, the map x → λF (x) is locally expanding. We show that the dynamics of this map define a representation of R d , d ≥ 2, as a union of simple curves γ : [0, ∞) → R d which tend to ∞ and whose interiors γ * = γ ((0, ∞)) are disjoint such that the union of all γ * has Hausdorff dimension 1.

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“…The motor for all the results that the Hausdorff dimension of dynamic rays is 1 is the following lemma of Karpińska [Ka99].…”

Section: Theorem 54 (The Dimension Paradox For Cosine Maps)mentioning

confidence: 99%

“…For details, see [Ka99,Sch07a]. The topological claims in Theorem 5.4 use standard contraction arguments using the hyperbolic metric in the finite postsingular set; see [Sch07b].…”

Section: Lemma 55 (The Parabola Lemma)mentioning

confidence: 99%

Dynamics of Entire Functions

Schleicher

2010

Lecture Notes in Mathematics

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Complex dynamics of iterated entire holomorphic functions is an active and exciting area of research. This manuscript collects known background in this field and describes several of the most active research areas within the dynamics of entire functions.Complex dynamics, in the sense of holomorphic iteration theory, has been a most active research area for the last three decades. A number of interesting developments have taken place during this time. After the foundational work by Fatou and Julia in the early 20th century, which developed much of the basic theory of iterated general rational (and also transcendental maps), the advent of computer graphics

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Hausdorff dimension of the hairs without endpoints for λ exp z

Cited by 54 publications

References 5 publications

Dynamic rays of bounded-type entire functions

Dynamic rays of bounded-type entire functions

Dynamics of a higher dimensional analog of the trigonometric functions

Dynamics of Entire Functions

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