Canonical compactifications for Markov shifts

2012 ◽  
Vol 33 (2) ◽  
pp. 441-454 ◽  
Author(s):  
DORIS FIEBIG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Complete Characterization ◽  
Markov Shift ◽  
Locally Compact ◽  
Markov Shifts ◽  
Countable State ◽  
State Mixing ◽  
The Given

AbstractWe give a complete characterization of the compact metric dynamical systems that appear as boundaries of the canonical compactification of a locally compact countable state mixing Markov shift. Consider such a compact metric dynamical system. Then there is a pair of non-conjugate Markov shifts with conjugate canonical compactifications, one of which has the given compact system as canonical boundary.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

2013 ◽  
Vol 23 (12) ◽  
pp. 1350196 ◽  
Author(s):  
JOSAPHAT R. R. GOUVEIA ◽  
FABÍOLO MORAES AMARAL ◽  
LUÍS F. C. ALBERTO
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Complete Characterization ◽  
Center Manifolds ◽  
The Stability ◽  
Hyperbolic Equilibrium ◽  
Autonomous Dynamical Systems

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

A cohomological characterization of amenable actions

1991 ◽  
Vol 110 (3) ◽  
pp. 491-504
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Dynamical Systems ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Classical Characterization

AbstractWe give a new characterization of amenability for dynamical systems, in cohomological terms, which generalizes the classical characterization of amenable locally compact groups stated by Johnson.

ABSTRACT ω-LIMIT SETS

2018 ◽  
Vol 83 (2) ◽  
pp. 477-495 ◽  
Author(s):  
WILL BRIAN
Keyword(s):  
Dynamical System ◽  
Hausdorff Space ◽  
Complete Characterization ◽  
Topological Dynamics ◽  
Limit Set ◽  
Limit Sets ◽  
Shift Map ◽  
Image Position ◽  
Continuous Surjection

AbstractThe shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

AN ITERATION METHOD FOR CHAOTIFYING AND CONTROLLING DYNAMICAL SYSTEMS

2002 ◽  
Vol 12 (05) ◽  
pp. 1173-1180 ◽  
Author(s):  
MAOYIN CHEN ◽  
ZHENGZHI HAN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Iteration Method ◽  
Chaotic Systems ◽  
Duffing Oscillator ◽  
Three Dimensional ◽  
Sufficient Condition ◽  
Chen System ◽  
The Given

This paper proposes a new iteration method for chaotifying and controlling dynamical systems. By applying this iteration method, the dimension of the given dynamical system can be reduced from to n to n-1. Moreover, the chaotified system is not necessarily Hurwitz stable originally. The iteration method is applied to three-dimensional systems for demonstration, for which a sufficient condition is obtained for chaotification. In addition, the iteration method can be used to control a class of chaotic systems. These results are illustrated via simulations on the Duffing oscillator and the Chen system.

scholarly journals Convex structures revisited

2015 ◽  
Vol 36 (5) ◽  
pp. 1596-1615 ◽  
Author(s):  
LIVIU PĂUNESCU
Keyword(s):  
Dynamical Systems ◽  
Extreme Points ◽  
Complete Characterization ◽  
Restriction Map ◽  
Convex Structure ◽  
Topological Dynamical Systems

We provide a complete characterization of extreme points of the space of sofic representations. We also show that the restriction map $\text{Sof}(G,P^{{\it\omega}})$ to $\text{Sof}(H,P^{{\it\omega}})$, where $H\subset G$, is not always surjective. The first part of the paper is a continuation of Păunescu [A convex structure on sofic embeddings. Ergod. Th. & Dynam. Sys.34(4) (2014), 1343–1352] and follows more closely the plan of Brown [Topological dynamical systems associated to $\text{II}_{1}$-factors. Adv. Math.227(4) (2011), 1665–1699].

A Problem of Gelfand on Rings of Operators and Dynamical Systems

1970 ◽  
Vol 22 (3) ◽  
pp. 514-517 ◽  
Author(s):  
Robert R. Kallman
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Borel Measure ◽  
Borel Function ◽  
Locally Compact Group ◽  
Countable Base ◽  
Borel Space ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
The Right

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:(1) s · g1g2 = (s · g1) · g2;(2) s · e = s;(3) (s, g) → s · g is a Borel function from S × G to S.If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].Consider the following general problem. Let (G, S, μ) be a dynamical system.

Expansive dynamics on locally compact groups

2020 ◽  
pp. 1-12
Author(s):  
BRUCE P. KITCHENS
Keyword(s):  
Dynamical System ◽  
Topological Group ◽  
Finite Number ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Coset Space ◽  
Locally Compact ◽  
Countable State ◽  
Full Shift ◽  
Totally Disconnected

Abstract Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

scholarly journals A Complete Characterization of Projectivity for Statistical Relational Models

2020 ◽  
Author(s):  
Manfred Jaeger ◽  
Oliver Schulte
Keyword(s):  
Latent Variable ◽  
Probability Distributions ◽  
Complete Characterization ◽  
Relational Models ◽  
Lifted Inference ◽  
Relational Structures ◽  
Statistical Frequency ◽  
Exchangeable Arrays ◽  
The Given

A generative probabilistic model for relational data consists of a family of probability distributions for relational structures over domains of different sizes. In most existing statistical relational learning (SRL) frameworks, these models are not projective in the sense that the marginal of the distribution for size-n structures on induced substructures of size k<n is equal to the given distribution for size-k structures. Projectivity is very beneficial in that it directly enables lifted inference and statistically consistent learning from sub-sampled relational structures. In earlier work some simple fragments of SRL languages have been identified that represent projective models. However, no complete characterization of, and representation framework for projective models has been given. In this paper we fill this gap: exploiting representation theorems for infinite exchangeable arrays we introduce a class of directed graphical latent variable models that precisely correspond to the class of projective relational models. As a by-product we also obtain a characterization for when a given distribution over size-k structures is the statistical frequency distribution of size-k substructures in much larger size-n structures. These results shed new light onto the old open problem of how to apply Halpern et al.'s ``random worlds approach'' for probabilistic inference to general relational signatures.

scholarly journals Parametric invariance and the Pioneer anomaly

2012 ◽  
Vol 90 (10) ◽  
pp. 931-937 ◽  
Author(s):  
Antonio F. Rañada ◽  
A. Tiemblo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Equations Of Motion ◽  
Classical Dynamics ◽  
Hamiltonian Form ◽  
Pioneer Anomaly ◽  
Dynamical Time ◽  
Complete Treatment

It is usually assumed that the t parameter in the equations of dynamics can be identified with the indication of the pointer of a clock. Things are not so simple, however. In fact, because the equations of motion can be written in terms of t but also in terms of t′ = f(t), f being any well-behaved function, any one of those infinite parametric times t′ is as good as the newtonian one to study classical dynamics in hamiltonian form. Here we show that, as a consequence of parametric invariance, one of the foundations of classical dynamics, the relation between the mathematical parametric time t in the equations of dynamics and the physical dynamical time σ that is measured with a particular clock (which is itself a dynamical system) requires the characterization of the clock that is used to achieve a complete treatment of dynamical systems. These two kinds of time, therefore, must be carefully distinguished. Furthermore, we show that not all the dynamical clock-times are necessarily equivalent and that the observational fingerprint of this nonequivalence has, curiously, the same form as that of the Pioneer anomaly. This suggests, therefore, that an acceleration to one another of the astronomical and the atomic times, tastr and tatom, can contribute to the total amount of the anomaly.

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