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} \} $.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Expansions in Complex Bases

2007 ◽  
Vol 50 (3) ◽  
pp. 399-408 ◽  
Author(s):  
Vilmos Komornik ◽  
Paola Loreti
Keyword(s):  
Number Theory ◽  
Dynamical Systems ◽  
Greedy Algorithm ◽  
Finite Automata ◽  
Complex Case ◽  
Seminal Paper

AbstractBeginning with a seminal paper of Rényi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daróczy and Kátai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

scholarly journals In Search of Chaos and Complexity of a Cognitive Language-Learning System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Gaofei Luo ◽  
Sayan Mukherjee
Keyword(s):  
Language Learning ◽  
System Parameter ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Learning System ◽  
Lyapunov Analysis ◽  
Nonlinear Structure ◽  
Dynamical Features ◽  
Power Noise

In this article, we investigate the long-term dynamics of a known cognitive-based language-learning system under the variation of a system parameter. Stability of the equilibrium points is studied. Period root to chaos is investigated by bifurcation analysis. A Lyapunov analysis is performed to verify the complex dynamics in the system. Existence of chaos is confirmed by 0-1 test. A noise-induced cognitive phenomenon is proposed under the effect of power noise. Chaotic and nonchaotic dynamics are explored in the noise-induced system. Furthermore, disorder as well as complexity, are investigated for both the systems using the concept of weighted recurrence. The whole analysis can be effective to understand the dynamical features and nonlinear structure of the cognitive language-learning model.

scholarly journals K-theory for Cuntz-Krieger algebras arising from real quadratic maps

2003 ◽  
Vol 2003 (34) ◽  
pp. 2139-2146 ◽  
Author(s):  
Nuno Martins ◽  
Ricardo Severino ◽  
J. Sousa Ramos
Keyword(s):  
Transition Matrix ◽  
Parameter Family ◽  
Quadratic Maps ◽  
Markov Transition Matrix ◽  
Kneading Sequence ◽  
Markov Transition ◽  
K Theory ◽  
The One ◽  
Real Quadratic

We compute theK-groups for the Cuntz-Krieger algebras𝒪A𝒦(fμ), whereA𝒦(fμ)is the Markov transition matrix arising from the kneading sequence𝒦(fμ)of the one-parameter family of real quadratic mapsfμ.

scholarly journals NONLINEAR AND COMPLEX DYNAMICS IN ECONOMICS

2014 ◽  
Vol 19 (8) ◽  
pp. 1749-1779 ◽  
Author(s):  
William A. Barnett ◽  
Apostolos Serletis ◽  
Demitre Serletis
Keyword(s):  
Dynamical Systems ◽  
Complex Dynamics ◽  
State Of The Art ◽  
Geometric Approach ◽  
Stable Region ◽  
Physical Sciences ◽  
Dynamical Complexity ◽  
Policy Relevance ◽  
Potential Applications ◽  
Transcritical Bifurcations

This paper is an up-to-date survey of the state of the art in dynamical systems theory relevant to high levels of dynamical complexity, characterizing chaos and near-chaos, as commonly found in the physical sciences. The paper also surveys applications in economics and finance. This survey does not include bifurcation analyses at lower levels of dynamical complexity, such as Hopf and transcritical bifurcations, which arise closer to the stable region of the parameter space. We discuss the geometric approach (based on the theory of differential/difference equations) to dynamical systems and make the basic notions of complexity, chaos, and other related concepts precise, having in mind their (actual or potential) applications to economically motivated questions. We also introduce specific applications in microeconomics, macroeconomics, and finance and discuss the policy relevance of chaos.

scholarly journals Shadowing and -limit sets of circular Julia sets

2013 ◽  
Vol 35 (4) ◽  
pp. 1045-1055 ◽  
Author(s):  
ANDREW D. BARWELL ◽  
JONATHAN MEDDAUGH ◽  
BRIAN E. RAINES
Keyword(s):  
Unit Circle ◽  
Complex Plane ◽  
Julia Sets ◽  
Closed Set ◽  
Quadratic Polynomials ◽  
Limit Sets ◽  
Kneading Sequence

AbstractIn this paper we consider quadratic polynomials on the complex plane${f}_{c} (z)= {z}^{2} + c$and their associated Julia sets,${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an$n$-tupling. In this case${J}_{c} $contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that${f}_{c} : {J}_{c} \rightarrow {J}_{c} $has shadowing, and we classify all$\omega $-limit sets for these maps by showing that a closed set$R\subseteq {J}_{c} $is internally chain transitive if, and only if, there is some$z\in {J}_{c} $with$\omega (z)= R$.

scholarly journals Enzyme-free nucleic acid dynamical systems

2017 ◽  
Author(s):  
Niranjan Srinivas ◽  
James Parkin ◽  
Georg Seelig ◽  
Erik Winfree ◽  
David Soloveichik
Keyword(s):  
Dynamical Systems ◽  
Programming Languages ◽  
Regulatory Networks ◽  
Molecular Mechanisms ◽  
Complex Dynamics ◽  
Molecular Systems ◽  
Synthetic Dna ◽  
Molecular Programming ◽  
And Control ◽  
Dna Components

Chemistries exhibiting complex dynamics—from inorganic oscillators to gene regulatory networks—have been long known but either cannot be reprogrammed at will, or rely on the sophisticated chemistry underlying the central dogma. Can simpler molecular mechanisms, designed from scratch, exhibit the same range of behaviors? Abstract coupled chemical reactions have been proposed as a programming language for complex dynamics, along with their systematic implementation using short synthetic DNA molecules. We developed this technology for dynamical systems, identifying critical design principles and codifying them into a compiler automating the design process. Using this approach, we built an oscillator containing only DNA components, establishing that Watson-Crick base pairing interactions alone suffice for arbitrarily complex dynamics. Our results argue that autonomous molecular systems that interact with and control their chemical environment can be designed via molecular programming languages.

Sign in / Sign up

Export Citation Format

Share Document