scholarly journals Concentration inequalities for sequential dynamical systems of the unit interval

2015 ◽  
Vol 36 (8) ◽  
pp. 2384-2407 ◽  
Author(s):  
ROMAIN AIMINO ◽  
JÉRÔME ROUSSEAU
Keyword(s):  
Dynamical Systems ◽  
Unit Interval ◽  
Concentration Inequalities ◽  
Concentration Inequality ◽  
Sequential Dynamical Systems

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm and we investigate several of its consequences. In particular, this covers compositions of$\unicode[STIX]{x1D6FD}$-transformations, with all$\unicode[STIX]{x1D6FD}$lying in a neighborhood of a fixed$\unicode[STIX]{x1D6FD}_{\star }>1$, and systems satisfying a covering-type assumption.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

Elements of a theory of simulation II: sequential dynamical systems

2000 ◽  
Vol 107 (2-3) ◽  
pp. 121-136 ◽  
Author(s):  
C.L. Barrett ◽  
H.S. Mortveit ◽  
C.M. Reidys
Keyword(s):  
Dynamical Systems ◽  
Sequential Dynamical Systems

NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

2004 ◽  
Vol 07 (03n04) ◽  
pp. 395-418
Author(s):  
H. S. MORTVEIT ◽  
C. M. REIDYS
Keyword(s):  
Dynamical Systems ◽  
Mutation Rate ◽  
Distance Measure ◽  
Neutral Evolution ◽  
Mutation Rates ◽  
Combinatorial Properties ◽  
Labeled Graph ◽  
Sequential Dynamical Systems ◽  
Natural Way ◽  
Independent Probability

In this paper we study the evolution of sequential dynamical systems [Formula: see text] as a result of the erroneous replication of the SDS words. An [Formula: see text] consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The [Formula: see text] over the word w and Y is the composed map: [Formula: see text]. The word w represents the genotype of the [Formula: see text] in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the [Formula: see text]. We introduce combinatorial properties of [Formula: see text] which allow us to construct a new distance measure [Formula: see text] for words. We show that [Formula: see text] captures the similarity of corresponding [Formula: see text]. We will use the distance measure [Formula: see text] to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.

scholarly journals Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Juan A. Aledo ◽  
Luis G. Diaz ◽  
Silvia Martinez ◽  
Jose C. Valverde
Keyword(s):  
Dynamical Systems ◽  
Upper Bound ◽  
Boolean Functions ◽  
Network Models ◽  
Garden Of Eden ◽  
Sequential Dynamical Systems

In this paper, we deal with one of the main computational questions in network models: the predecessor-existence problems. In particular, we solve algebraically such problems in sequential dynamical systems on maxterm and minterm Boolean functions. We also provide a description of the Garden-of-Eden configurations of any system, giving the best upper bound for the number of Garden-of-Eden points.

Neutral networks: a combinatorial perspective

1999 ◽  
Vol 02 (03) ◽  
pp. 283-301 ◽  
Author(s):  
Stephan Kopp ◽  
Christian M. Reidys
Keyword(s):  
Combinatorial Optimization ◽  
Dynamical Systems ◽  
Computer Simulations ◽  
Evolutionary Change ◽  
Rna Molecules ◽  
New Class ◽  
Neutral Networks ◽  
Threshold Phenomenon ◽  
Theoretical Investigations ◽  
Sequential Dynamical Systems

The existence of neutral networks in genotype-phenotype maps has provided significant insight in theoretical investigations of evolutionary change and combinatorial optimization. In this paper we will consider neutral networks of two particular genotype-phenotype maps from a combinatorial perspective. The first map occurs in the context of folding RNA molecules into their secondary structures and the second map occurs in the study of sequential dynamical systems, a new class of dynamical systems designed for the analysis of computer simulations. We will prove basic properties of neutral nets and present an error threshold phenomenon for evolving populations of simulation schedules.

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