scholarly journals Multiplicative combinatorial properties of return time sets in minimal dynamical systems

2019 ◽  
Vol 39 (10) ◽  
pp. 5891-5921
Author(s):  
Daniel Glasscock ◽  
◽  
Andreas Koutsogiannis ◽  
Florian Karl Richter ◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Return Time ◽  
Combinatorial Properties ◽  
Minimal Dynamical Systems

scholarly journals Embedding minimal dynamical systems into Hilbert cubes

2020 ◽  
Vol 221 (1) ◽  
pp. 113-166 ◽  
Author(s):  
Yonatan Gutman ◽  
Masaki Tsukamoto
Keyword(s):  
Dynamical Systems ◽  
Minimal Dynamical Systems

scholarly journals UHF-slicing and classification of nuclear C*-algebras

2014 ◽  
Vol 06 (04) ◽  
pp. 465-540 ◽  
Author(s):  
Karen R. Strung ◽  
Wilhelm Winter
Keyword(s):  
Dynamical Systems ◽  
Classification Theorem ◽  
Discrete Version ◽  
C Algebras ◽  
Minimal Dynamical Systems ◽  
Tracial States

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

scholarly journals Skew products and minimal dynamical systems on separable Hilbert manifolds

1984 ◽  
Vol 4 (2) ◽  
pp. 213-224 ◽  
Author(s):  
A. Fathi
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Separable Hilbert Space ◽  
Countable Group ◽  
Locally Compact ◽  
Connected Manifold ◽  
Skew Products ◽  
Minimal Dynamical Systems

AbstractWe prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.

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 A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

2021 ◽  
Author(s):  
Maximilian Engel ◽  
Christian Kuehn
Keyword(s):  
Dynamical Systems ◽  
Limit Cycle ◽  
Level Sets ◽  
Return Time ◽  
Random Dynamical Systems ◽  
Expected Return ◽  
Stable Manifolds ◽  
Kolmogorov Operator ◽  
Definition Of ◽  
Stochastic Oscillations

AbstractFor an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

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