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).

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 Number-theoretic positive entropy shifts with small centralizer and large normalizer

2020 ◽  
pp. 1-26
Author(s):  
M. BAAKE ◽  
Á. BUSTOS ◽  
C. HUCK ◽  
M. LEMAŃCZYK ◽  
A. NICKEL
Keyword(s):  
Dynamical System ◽  
Number Fields ◽  
Lattice Points ◽  
Topological Invariants ◽  
Positive Entropy ◽  
Quadratic Number ◽  
Positive Topological Entropy ◽  
Higher Dimensional ◽  
Real Quadratic ◽  
Shift Dynamical System

Abstract Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

The role of the principal part in factorizing block maps

1984 ◽  
Vol 96 (2) ◽  
pp. 223-235
Author(s):  
Frank Rhodes
Keyword(s):  
Dynamical System ◽  
Computational Complexity ◽  
Principal Part ◽  
Principal Vector ◽  
Close Relationship ◽  
Principal Parts ◽  
Weaker Conditions ◽  
Role Of The Principal ◽  
Shift Dynamical System

An n-block is a sequence b1 … bn where bi ε {0,1} for 1 ≤ i ≤ n, and an n-block map is a function from the set of n-blocks to the set {0,1}. Block maps can be used to study endomorphisms of the shift dynamical system [7] and shift register sequences [4]. Composition of endomorphisms and cascading of shift registers [6] can be studied via composition of block maps. If fog is a given block map which is linear in the first variable and if g is also given, then f is unique and can be determined. If fog and f are given then g is unique and can be determined, at least up to a constant [10]. However, little is known about the level of computational complexity of the general task of searching for factors of a given block map. In this paper I identify a substantial class of maps which are linear in the first variable but not linear, for which there are effective methods of searching for factors. The methods are based on the presentation of block maps via successive principal parts, introduced in [9]. There I define the principal vector of a block map. One of the key results gives conditions under which there is a particularly close relationship between the principal vectors of two block maps f and g and their composition fog. In the second section of this paper I show that the same relationship holds under weaker conditions.

scholarly journals The number of ergodic measures for transitive subshifts under the regular bispecial condition

2020 ◽  
pp. 1-55
Author(s):  
MICHAEL DAMRON ◽  
JON FICKENSCHER
Keyword(s):  
Dynamical System ◽  
Shift Operator ◽  
Interval Exchange ◽  
Combinatorial Proof ◽  
Interval Exchange Transformations ◽  
Left Shift ◽  
Ergodic Measures ◽  
Finite Set ◽  
Limiting Value ◽  
Shift Dynamical System

Abstract If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$ , along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition (‘regular bispecial condition’) on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $({K+1})/{2}$ ergodic measures, where K is the limiting value of $p(n+1)-p(n)$ , and p is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from 1984, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.

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 Counting Closed Orbits for the Dyck Shift

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fahad Alsharari ◽  
Mohd Salmi Md. Noorani ◽  
Habibulla Akhadkulov
Keyword(s):  
Dynamical System ◽  
Direct Methods ◽  
Theoretical Discussion ◽  
Infinite Type ◽  
Closed Orbits ◽  
Shift Dynamical System

The prime orbit theorem and Mertens’ theorem are proved for a shift dynamical system of infinite type called the Dyck shift. Different and more direct methods are used in the proof without any complicated theoretical discussion.

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