On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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 $f$.

Zero-dimensional covers of finite dimensional dynamical systems

1995 ◽  
Vol 15 (5) ◽  
pp. 939-950 ◽  
Author(s):  
John Kulesza
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

scholarly journals On the Rank and Periodic Rank of Finite Dynamical Systems

10.37236/7017 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Maximilien Gadouleau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Boolean Networks ◽  
Finite Alphabet ◽  
Interaction Graph ◽  
Maximum Rank ◽  
Local Functions ◽  
Finite Dynamical Systems ◽  
Finite Dynamical System

A finite dynamical system is a function $f : A^n \to A^n$ where $A$ is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.

THE DYNAMICAL SYSTEMS ASSOCIATED WITH CHEBYSHEV POLYNOMIALS IN TWO VARIABLES

1996 ◽  
Vol 06 (12b) ◽  
pp. 2611-2618 ◽  
Author(s):  
KEISUKE UCHIMURA
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Complex Variable ◽  
Chebyshev Polynomials ◽  
Complex Conjugate ◽  
Closed Domain ◽  
Shift Dynamical System

The dynamical system given by [Formula: see text] is considered, where z is a complex variable and [Formula: see text] denotes the complex conjugate of it. The function F2(z) is related to Chebyshev polynomials in two variables. We show the chaoticity of this dynamical system on some closed domain and relations between the dynamics and a shift dynamical system. Besides we show that the dynamical system is related to the Sierpinsky gasket.

scholarly journals Dynamical properties of spatial discretizations of a generic homeomorphism

2014 ◽  
Vol 35 (5) ◽  
pp. 1474-1523 ◽  
Author(s):  
PIERRE-ANTOINE GUIHÉNEUF
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Numerical Study ◽  
Periodic Points ◽  
Dynamical Behaviour ◽  
Compact Manifolds ◽  
Spatial Discretization ◽  
Dynamical Features ◽  
Dynamical Properties

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

scholarly journals Periodic points and transitivity on dendrites

2016 ◽  
Vol 37 (7) ◽  
pp. 2017-2033 ◽  
Author(s):  
GERARDO ACOSTA ◽  
RODRIGO HERNÁNDEZ-GUTIÉRREZ ◽  
ISSAM NAGHMOUCHI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
Dense Orbit ◽  
Topological Graphs ◽  
Periodic Decomposition ◽  
Dense Set ◽  
Compact Subsets

We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It is also possible that a dynamical system is transitive but there is a unique periodic point which, in fact, is the unique fixed point. We also prove that on almost meshed continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all non-empty compact subsets.

scholarly journals On Transitive Points in a Generalized Shift Dynamical System

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Bahman Taherkhani ◽  
Fatemah Ayatollah Zadeh Shirazi
Keyword(s):  
Dynamical System ◽  
Transitive Point ◽  
Generalized Shift ◽  
Shift Dynamical System

Considering point transitive generalized shift dynamical system (XΓ,σφ) for discrete X with at least two elements and infinite Γ, we prove that X is countable and Γ has at most 2ℵ0 elements. Then, we find a transitive point of the dynamical system (NN×Z,στ) for τ:N×Z→N×Z with τ(n,m)=(n,m+1) and show that point transitive (XΓ,σφ), for infinite countable Γ, is a factor of (NN×Z,στ).

scholarly journals Properties of invariant measures in dynamical systems with the shadowing property

2017 ◽  
Vol 38 (6) ◽  
pp. 2257-2294 ◽  
Author(s):  
JIAN LI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Entropy Function ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Ergodic Measures ◽  
One To One ◽  
Transitive System

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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