scholarly journals Bernoulli measure of complex admissible kneading sequences

2013 ◽  
Vol 33 (3) ◽  
pp. 821-830 ◽  
Author(s):  
HENK BRUIN ◽  
DIERK SCHLEICHER
Keyword(s):  
Dynamical Systems ◽  
Complex Dynamics ◽  
Infinite Sequence ◽  
Complex Case ◽  
Simple Complex ◽  
Quadratic Polynomials ◽  
Kneading Sequence ◽  
Bernoulli Measure ◽  
Dynamical Features ◽  
Real Quadratic

AbstractIterated quadratic polynomials give rise to a rich collection of different dynamical systems that are parametrized by a simple complex parameter $c$. The different dynamical features are encoded by the kneading sequence, which is an infinite sequence over $\{ \mathtt{0} , \mathtt{1} \} $. Not every such sequence actually occurs in complex dynamics. The set of admissible kneading sequences was described by Milnor and Thurston for real quadratic polynomials, and by the authors in the complex case. We prove that the set of admissible kneading sequences has positive Bernoulli measure within the set of sequences over $\{ \mathtt{0} , \mathtt{1} \} $.

Dynamics of quasiregular mappings with constant complex dilatation

2014 ◽  
Vol 36 (2) ◽  
pp. 514-549 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
ROB FRYER
Keyword(s):  
Complex Dynamics ◽  
Quasiregular Mapping ◽  
Quadratic Polynomial ◽  
Complex Dilatation ◽  
Quasiregular Mappings ◽  
Quadratic Polynomials ◽  
Linear Map ◽  
External Rays ◽  
Constant Complex

The study of quadratic polynomials is a foundational part of modern complex dynamics. In this article, we study quasiregular counterparts to these in the plane. More specifically, let $h:\mathbb{C}\rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear map and consider the quasiregular mapping $H=g\circ h$, where $g$ is a quadratic polynomial. By studying $H$ and via the Böttcher-type coordinate constructed in A. Fletcher and R. Fryer [On Böttcher coordinates and quasiregular maps. Contemp. Math.575 (2012), 53–76], we are able to obtain results on the dynamics of any degree-two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. We also show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.

Frontiers in Complex Dynamics

2014 ◽  
Author(s):  
Araceli Bonifant ◽  
Misha Lyubich ◽  
Scott Sutherland
Keyword(s):  
Historical Perspective ◽  
Complex Dynamics ◽  
Short Paper ◽  
Combinatorial Group Theory ◽  
Entropy Theory ◽  
Holomorphic Dynamics ◽  
Eightieth Birthday ◽  
Several Variables ◽  
Fields Medal ◽  
Combinatorial Group

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

An Alternative Approach To User Fee Policy In The Primary Health Care Setting Of Bangladesh.

2020 ◽  
Author(s):  
Nazneen Akhter
Keyword(s):  
Health Care ◽  
Primary Health Care ◽  
Complex Dynamics ◽  
Quality Care ◽  
Secondary Data ◽  
User Fees ◽  
Primary Health ◽  
Alternative Approach ◽  
Benefit Principle ◽  
User Fee

The concept of ascribing user fee in health care settings always remained a policy struggle and countries experienced different learning in this regards while implementing user fee at different tiers of health settings. The most exquisite learning among the many country specific evidences related to user fee are the match and mismatch between the equity principle and benefit principle while considering the client perspective. There is an added dimension of quality care which also add more complex dynamics into this concept since the quality care consideration has a double edged perspective both for clients and providers, where which one will get superiority over whom is a great question in health care, especially in the Primary Health care (PHC) of the country. In this reality the appropriate implementation guideline, followed by an appropriate practice of the administrative and management both service oriented and financial are of great importance in this user fee implementation consideration which always remained a challenge in the health care specially in remote care of PHC. This paper attempted a secondary data searching and scoping the available documents of Bangladesh and across the world to find an alternative approach to user fees policy where equity and benefit principle and quality - these three have to be placed in a well-constructed triad in PHC implementation which has been recommended as an alternative policy imperative in approaching user fees for Bangladesh PHC settings.

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