scholarly journals Dense set of negative Schwarzian maps whose critical points have minimal limit sets

1998 ◽  
Vol 4 (1) ◽  
pp. 141-158
Author(s):  
Alexander Blokh ◽  
◽  
Michał Misiurewicz ◽  
Keyword(s):  
Critical Points ◽  
Limit Sets ◽  
Dense Set

Meromorphic multifunctions in complex dynamics

1992 ◽  
Vol 12 (1) ◽  
pp. 39-52 ◽  
Author(s):  
L. Baribeau ◽  
T. J. Ransford
Keyword(s):  
Critical Points ◽  
Complex Dynamics ◽  
Julia Set ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Analytic Family ◽  
Limit Sets ◽  
Upper Semicontinuous ◽  
Open Set ◽  
Similar Vein

AbstractLet {RA} be an analytic family of rational maps and denote by j(λ) the Julia set of Rλ. We prove that the upper semicontinuous regularization j(λ) of j(λ) (which coincides with j(λ) for all λ in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.

Typical limit sets of critical points for smooth interval maps

2000 ◽  
Vol 20 (1) ◽  
pp. 15-45 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
MICHAŁ MISIUREWICZ
Keyword(s):  
Critical Points ◽  
Limit Sets ◽  
Interval Maps ◽  
Class C

We prove that interval maps for which $\omega$-limit sets of all critical points are minimal are dense in the space of all interval maps of class $C^2$.

Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows

2009 ◽  
Vol 52 (2) ◽  
pp. 315-320
Author(s):  
Taishan Yi ◽  
Xingfu Zou
Keyword(s):  
Protein Synthesis ◽  
State Space ◽  
The State ◽  
Limit Set ◽  
Limit Sets ◽  
Realistic Significance ◽  
Dense Set

AbstractBy employing the limit set dichotomy for essentially strongly order-preserving semiflows and the assumption that limit sets have infima and suprema in the state space, we prove a generic quasi-convergence principle implying the existence of an open and dense set of stable quasi-convergent points. We also apply this generic quasi-convergence principle to a model for biochemical feedback in protein synthesis and obtain some results about the model which are of theoretical and realistic significance.

scholarly journals Recurrent critical points and typical limit sets of rational maps

1999 ◽  
Vol 127 (4) ◽  
pp. 1215-1220 ◽  
Author(s):  
Alexander M. Blokh ◽  
John C. Mayer ◽  
Lex G. Oversteegen
Keyword(s):  
Critical Points ◽  
Rational Maps ◽  
Limit Sets

scholarly journals Porosity in Conformal Dynamical Systems

2021 ◽  
pp. 1-69
Author(s):  
VASILEIOS CHOUSIONIS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Continued Fractions ◽  
Julia Set ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Markov Systems ◽  
Other Hand ◽  
Almost Everywhere ◽  
Complex Continued Fractions ◽  
Dense Set

Abstract In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction. We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere. We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

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.

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