A result about topologically transitive set

2016 ◽  
Author(s):  
Chao Liang
Keyword(s):  
Topologically Transitive ◽  
Transitive Set

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

2011 ◽  
Vol 13 (02) ◽  
pp. 191-211 ◽  
Author(s):  
D. CARRASCO-OLIVERA ◽  
C. A. MORALES ◽  
B. SAN MARTÍN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Manifold With Boundary ◽  
Transitive Set ◽  
Singular Cycle ◽  
Hyperbolic Equilibrium

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

scholarly journals Characterization of topologically transitive attractors for analytic plane flows

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Martin Shoptrajanov ◽  
Nikita Shekutkovski
Keyword(s):  
Necessary Conditions ◽  
Topological Characterization ◽  
Plane Flow ◽  
Limit Sets ◽  
Topologically Transitive ◽  
Analytic Plane

We give necessary conditions for a set to be topologically transitive attractor of an analytic plane flow using topological characterization of ?-limit sets and the concept of upper semi-continuity of multi valued maps.

scholarly journals A novel route to cyclic dominance in voluntary social dilemmas

2020 ◽  
Vol 17 (164) ◽  
pp. 20190789 ◽  
Author(s):  
Hao Guo ◽  
Zhao Song ◽  
Sunčana Geček ◽  
Xuelong Li ◽  
Marko Jusup ◽  
...  
Keyword(s):  
Evolutionary Dynamics ◽  
Social Dilemmas ◽  
Modern Human ◽  
Voluntary Participation ◽  
Risk Averse ◽  
Bacterial Strains ◽  
Adverse Conditions ◽  
Tit For Tat ◽  
Transitive Set ◽  
Set Up

Cooperation is the backbone of modern human societies, making it a priority to understand how successful cooperation-sustaining mechanisms operate. Cyclic dominance, a non-transitive set-up comprising at least three strategies wherein the first strategy overrules the second, which overrules the third, which, in turn, overrules the first strategy, is known to maintain biodiversity, drive competition between bacterial strains, and preserve cooperation in social dilemmas. Here, we present a novel route to cyclic dominance in voluntary social dilemmas by adding to the traditional mix of cooperators, defectors and loners, a fourth player type, risk-averse hedgers, who enact tit-for-tat upon paying a hedging cost to avoid being exploited. When this cost is sufficiently small, cooperators, defectors and hedgers enter a loop of cyclic dominance that preserves cooperation even under the most adverse conditions. By contrast, when the hedging cost is large, hedgers disappear, consequently reverting to the traditional interplay of cooperators, defectors, and loners. In the interim region of hedging costs, complex evolutionary dynamics ensues, prompting transitions between states with two, three or four competing strategies. Our results thus reveal that voluntary participation is but one pathway to sustained cooperation via cyclic dominance.

Remarks on Livšic' theory for nonabelian cocycles

1999 ◽  
Vol 19 (3) ◽  
pp. 703-721 ◽  
Author(s):  
KLAUS SCHMIDT
Keyword(s):  
Dynamical System ◽  
Homomorphic Image ◽  
Periodic Points ◽  
Skew Product ◽  
Topologically Transitive ◽  
Polish Group ◽  
Bounded Distortion

Let $(X,\phi)$ be a hyperbolic dynamical system and let $(G,\delta)$ be a Polish group. Motivated by Nicol and Pollicott, and then by Parry we study conditions under which two Hölder maps $f,g: X\longrightarrow G$ are Hölder cohomologous.In the context of Nicol and Pollicott we show that if $f$ and $g$ are measurably cohomologous and the distortion of the metric $\delta $ by the cocycles defined by $f$ and $g$ is bounded in an appropriate sense, then $f$ and $g$ are Hölder cohomologous.Two further results extend the main theorems recently presented by Parry. Under the hypothesis of bounded distortion we show that, if $f$ and $g$ give equal weight to all periodic points of $\phi $, then $f$ and $g$ are Hölder cohomologous. If the metric $\delta $ is bi-invariant, and if the skew-product $\phi _f$ defined by $f$ is topologically transitive, then conjugacy of weights implies that $g$ is Hölder conjugate to $\alpha \cdot f$ for some isometric automorphism $\alpha $ of $G$. The weaker condition that $g$-weights of periodic points are close to the identity whenever their $f$-weights are close to the identity implies that $g$ is continuously cohomologous to a homomorphic image of $f$.

Topological Transitivity on the Torus

1994 ◽  
Vol 37 (4) ◽  
pp. 549-551 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Vector Field ◽  
Finite Number ◽  
Fixed Points ◽  
Analytic Vector ◽  
Topological Transitivity ◽  
Topologically Transitive ◽  
Transitive Flow ◽  
Real Analytic ◽  
Analytic Vector Field

AbstractT. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.

scholarly journals ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

2001 ◽  
Vol 04 (04) ◽  
pp. 569-577 ◽  
Author(s):  
MATTHIAS GUNDLACH ◽  
ANDREI KHRENNIKOV ◽  
KARL-OLOF LINDAHL
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Unit Circle ◽  
Complex Plane ◽  
Haar Measure ◽  
Analogous Result ◽  
Topologically Transitive ◽  
Ergodic Behavior ◽  
Adic Case

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

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