Uniformly perfect sets and the Poincar� metric

1979 ◽  
Vol 32 (1) ◽  
pp. 192-199 ◽  
Author(s):  
Ch. Pommerenke
Keyword(s):  
Uniformly Perfect

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s

2000 ◽  
Vol 128 (9) ◽  
pp. 2569-2575 ◽  
Author(s):  
Rich Stankewitz
Keyword(s):  
Kleinian Groups ◽  
Uniformly Perfect

scholarly journals Uniformization of Cantor sets with bounded geometry

2021 ◽  
Vol 25 (5) ◽  
pp. 88-103
Author(s):  
Vyron Vellis
Keyword(s):  
Compact Subset ◽  
Higher Dimensions ◽  
Cantor Set ◽  
Cantor Sets ◽  
Content Type ◽  
Bounded Geometry ◽  
Uniformly Perfect ◽  
Disconnected Sets

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} .

scholarly journals LOWER ASSOUAD-TYPE DIMENSIONS OF UNIFORMLY PERFECT SETS IN DOUBLING METRIC SPACES

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050039
Author(s):  
HAIPENG CHEN ◽  
MIN WU ◽  
YUANYANG CHANG
Keyword(s):  
Metric Spaces ◽  
The Other ◽  
Equivalent Definition ◽  
Box Counting ◽  
Assouad Dimension ◽  
Uniformly Perfect ◽  
Box Counting Dimension ◽  
Definition Of ◽  
The Relationship ◽  
Variational Result

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as [Formula: see text] tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as [Formula: see text] tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.

scholarly journals The limit sets of Schottky quasiconformal groups are uniformly perfect

2005 ◽  
Vol 357 (6) ◽  
pp. 2119-2132
Author(s):  
Xiaosheng Li
Keyword(s):  
Limit Sets ◽  
Uniformly Perfect

scholarly journals Julia sets of uniformly quasiregular mappings are uniformly perfect

2011 ◽  
Vol 151 (3) ◽  
pp. 541-550 ◽  
Author(s):  
ALASTAIR N. FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Hausdorff Dimension ◽  
Analogous Result ◽  
Julia Set ◽  
Julia Sets ◽  
Quasiregular Mapping ◽  
Higher Dimensions ◽  
Ring Domain ◽  
Rational Map ◽  
Quasiregular Mappings ◽  
Uniformly Perfect

AbstractIt is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

scholarly journals ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

2011 ◽  
Vol 91 (3) ◽  
pp. 289-311 ◽  
Author(s):  
WALTER BERGWEILER ◽  
JIAN-HUA ZHENG
Keyword(s):  
Entire Function ◽  
General Criterion ◽  
Multiply Connected ◽  
Uniformly Perfect ◽  
Wandering Domain ◽  
Uniform Perfectness ◽  
Wandering Domains

AbstractWe investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

scholarly journals Lower Assouad Dimension of Measures and Regularity

2019 ◽  
pp. 1-37 ◽  
Author(s):  
KATHRYN E. HARE ◽  
SASCHA TROSCHEIT
Keyword(s):  
Lower Bounds ◽  
Open Set Condition ◽  
Local Behaviour ◽  
Pisot Numbers ◽  
Open Set ◽  
Regularity Properties ◽  
Assouad Dimension ◽  
Uniformly Perfect ◽  
Bernoulli Convolutions ◽  
Self Similar

Abstract In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

Sign in / Sign up

Export Citation Format

Share Document