On multifractal spectrum of quasiconformal mappings

Hitruhin, Lauri

doi:10.5186/aasfm.2016.4138

Cited by 6 publications

(14 citation statements)

References 4 publications

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“…Then, using the definition of the mapping f and the definition (6.4) of the radial map ψ B,K , we will estimate the argument (6.8) | arg(f (z + λ z,n ) − f (z))| as in [7]. Namely, we first sum up the rotation coming from crossing the annuli A j,ij , which can be calculated from (6.4) to be of order K n for every j ∈ {1, 2, .…”

Section: Sharpness Of Theorems 11 and 12 And An Applicationmentioning

confidence: 99%

Joint rotational and stretching multifractal spectra of mappings with integrable distortion

Hitruhin

2019

Rev. Mat. Iberoam.

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We establish bounds for both the stretching and the rotational multifractal spectra of planar homeomorphic mappings with p-integrable distortion. Moreover, we show that these bounds are sharp by presenting extremal examples of homeomorphic mappings with p-integrable distortion that attain the optimal stretching and rotation simultaneously, which additionally leads to the sharp bound for the joint rotational and stretching multifractal spectra. Finally, we will use the stretching multifractal spectra to obtain the sharp area contraction for homeomorphic mappings with p-integrable distortion.

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Section: Sharpness Of Theorems 11 and 12 And An Applicationmentioning

confidence: 99%

Joint rotational and stretching multifractal spectra of mappings with integrable distortion

Hitruhin

2019

Rev. Mat. Iberoam.

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“…Moreover, the limiting function of the construction is K-quasiconformal as long as the parameters (α, γ) are chosen to allow this. In particular, following [5], we can choose α, γ to be any pair for which F K (α, γ) = 0.…”

Section: Dimension Zeromentioning

confidence: 99%

“…Following the argument in [5], we end up with the result that the total rotation as we move from ∞ to a disk at scale n is…”

Section: Dimension Greater Than Zeromentioning

confidence: 99%

“…In the direction of lower bounds, that paper gives constructions to attain all dimensions below the bound F K (α, γ). Hitruhin improved this in [5] to give examples of quasiconformal maps whose stretching and rotation sets have positive and finite Hausdorff measure at the critical dimension. That paper used a Cantor set construction from [8] to prove this; the work gives a construction of a quasiconformal map whose distortion of a family of disks is completely understood.…”

Section: Introductionmentioning

confidence: 99%

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Stretching and rotation sets of quasiconformal mappings

Bongers¹

2019

Ann. Fenn. Math.

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Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.

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“…Lately there has been a significant upsurge of interest in understanding the pointwise rotational properties of planar homeomorphisms of finite distortion, along with the spiraling rate of these maps, see [2,4,7,8,9,10]. To be precise, given such a homeomorphism f : C → C normalized by f (0) = 0 and f (1) = 1, one is interested in the maximal growth of | arg(f (r))| as r → 0.…”

Section: Introductionmentioning

confidence: 99%