On PZ type Siegel disks of the sine family

2014 ◽  
Vol 36 (3) ◽  
pp. 973-1006
Author(s):  
GAOFEI ZHANG
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Siegel Disk ◽  
Sine Family ◽  
Rotation Numbers

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.

scholarly journals Bounded-type Siegel disks of a one-dimensional family of entire functions

2009 ◽  
Vol 29 (1) ◽  
pp. 137-164 ◽  
Author(s):  
LINDA KEEN ◽  
GAOFEI ZHANG
Keyword(s):  
Fixed Point ◽  
Critical Points ◽  
Entire Functions ◽  
Irrational Number ◽  
Siegel Disk ◽  
One Dimensional ◽  
The Family

AbstractLet 0<θ<1 be an irrational number of bounded type. We prove that for any map in the family (e2πiθz+αz2)ez, α∈ℂ, the boundary of the Siegel disk, with fixed point at the origin, is a quasi-circle passing through one or both of the critical points.

Rational maps whose Fatou components are Jordan domains

1996 ◽  
Vol 16 (6) ◽  
pp. 1323-1343 ◽  
Author(s):  
Kevin M. Pilgrim
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Fatou Component ◽  
Jordan Curves ◽  
Fatou Components

AbstractWe prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

Critical points of metallic plasmas and electrolytes

2000 ◽  
Vol 10 (PR5) ◽  
pp. Pr5-373-Pr5-376 ◽  
Author(s):  
A. A. Likalter ◽  
H. Schneidenbach
Keyword(s):  
Critical Points

Application of graphical analysis of taxonomical structure of diatom complexes in identification of critical conditions of the Neopleistocene lakes (on the example of Bibirevo section)

2019 ◽  
pp. 78-87
Author(s):  
Elena V. Bespalova
Keyword(s):  
Lake Sediments ◽  
Critical Points ◽  
Graphical Analysis ◽  
Volga Region ◽  
Critical Conditions ◽  
Ancient Lake ◽  
Diatom Complexes

Ancient lake sediments of Bibirevo section in the Yaroslavl and Kostroma Volga region are studied by means of graphical analysis of taxonomical structure of diatom complexes. This method allowed to record critical points (change of areas of stability) in the development of a Neopleistocene lake during the transition from stage to stage, as well as from phase to phase.

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