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European Journal of Applied Sciences – Vol. 9, No. 5
Publication Date: October 25, 2021
DOI:10.14738/aivp.95.10762. Biswas, H. R., Islam, S. (2021). Topological Conjugacy and Symbolic Dynamics of the One Dimensional Map. European Journal of
Applied Sciences, 9(5). 44-55.
Services for Science and Education – United Kingdom
Topological Conjugacy and Symbolic Dynamics of the One
Dimensional Map
Hena Rani Biswas
Department of Mathematics, University of Barishal
Barishal-8200, Bangladesh
Md. Shahidul Islam
Department of Mathematics, University of Dhaka
Dhaka-1000, Bangladesh
ABSTRACT
In the study of nonlinear physical systems, one encounters possibly random or
chaotic behavior, although the techniques may be perfectly deterministic.
Topological conjugacy is essential in the study of iterated functions and, more
generally, dynamical systems. Our goal is to show that conjugacy (topological) and
symbolic dynamics are a distinguished combination of tools of dynamical systems.
This paper investigates conjugacy between different one-dimensional maps and
discusses semi-conjugacy between doubling maps and shift maps. Finally, we
discuss employing techniques from symbolic dynamics to one-dimensional maps,
which means how extended dynamics work for one dimensional map.
Keywords: Symbolic dynamics, Symbol space, Shift amp, Cantor set, Topological
conjugacy, semi-conjugacy.
INTRODUCTION
The Topological conjugacy feature has an essential role in studying the chaotic behavior of a
map. With the help of this feature, we can explore the chaotic significance by comparing one
map with another map. Topological conjugacy has such importance as it can protect many
topological dynamical properties. In the discussion of any chaotic dynamical system,
topological conjugacy between maps is a very powerful tool. Topological conjugacy is a
significant concept in the knowledge of dynamical systems. It is an essential tool that makes
predictions about a dynamical system's behavior by comparing it with another dynamical
system whose specific properties are known. We get the basic idea about symbolic dynamics
from many papers and books. There are some interesting applications of a symbolic dynamical
system. Such as the symbolic representation of the Cantor set. From the article of H.R. Biswas
and Monirul Islam [3], we get the basic idea of shift maps and their chaotic properties. Ju H.,
Shao H., Choe Y., and Shi Y. [11] narrated conditions for maps to be conjugate (topologically) or
semi-conjugate to subshifts of finite type and inference of chaos.
The symbolic dynamical systems form a significant category of dynamical systems. In such a
place, we may consider a shift operator that shifts a sequence one symbol left. J. Hadamar [1]
gave the example of the first application of symbolic dynamics. He used trajectory coding for
the representation of the universal behavior of geodesics on surfaces with negative curvature.
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Biswas, H. R., Islam, S. (2021). Topological Conjugacy and Symbolic Dynamics of the One Dimensional Map. European Journal of Applied Sciences,
9(5). 44-55.
URL: http://dx.doi.org/10.14738/aivp.95.10762
C. Hsu enlarged the cell-to-cell mapping method. Brin [7] discussed cells of a given partition
and their images under the action of a system. G. Osipenko [13] introduced the method of a
symbolic image in which way an oriented graph represents the transitions of a trajectory of
partition elements. Its adjacency matrix defines the Topological Markov chain, and symbol
sequences correspond to paths on the graph. This method is effectively applied to the cost of
production of invariant sets and Morse spectrum. Symbolic dynamics attempts to answer how
much actuality about a dynamical system can be drawn from a data sequence produced by
measurements of the system. Symbolic dynamics gives a method for converting real system
trajectories into symbol sequences and then answering how much about the underlying system
can be deduced from these sequences.
Symbolic dynamics hope that one may convert a dynamical system into an extended technique,
study the simplified dynamics in sequences space, and then take the results back to the state- space of the original system. Moreover, there are several works on � symbolic dynamics where
dynamics are represented by maps on symbol spaces.
In Section 2, we describe the mathematical preliminaries, which are requirements for the
subsequent chapters. In Section 3, we have proved that ∃ a conjugacy for shift map and tent
map. Also, in this section, we observed that a conjugacy between doubling map and shift map.
In Section 4, we have discussed symbolic dynamics for one dimensional map.
MATHEMATICAL PRELIMINARIES
The symbol space we use here has a metric structure, which is defined in a natural way. So the
study can be related to the metric space with the advantage that we can use the standard
concepts of dynamical systems available therein. Symbolic dynamical systems are suitable for
exalted generalization and abstraction of the original dynamical systems based on the
topological conjugacy between the continuous evolution of the dynamical systems. When the
original dynamical systems are difficult to be resolved, symbolic dynamics can provide a
promising direction. By symbolic dynamical systems, we mean here the space of sequences
∑! ={�: � =(�"�#�! . . . . . .), �& = � or �& = 1 for all �} along with the shift map defined on it.
Symbolic dynamics are trembling with maps on sets. We now introduce some basic definitions,
lemma, and theorem, which are essential to this paper.
Definition 2.1:
A map �: � → � is a conjugacy (topological) from �#: � → � to �!: � → � if
(i) k is a homomorphism from X to Y and
(ii) ��# = �!� or equivalently �! = ��#�'# or � = �'#�!� .
If the map � is continuous onto, they are said to be semi-conjugate (topologically). If �! is
topologically semi-conjugated to �# , �# having a dense set of periodic points and being
topologically transitive, then �! is a topologically transitive and has a dense set of periodic
points.
Semi-conjugacy and conjugacy are more commonly related to a commuted diagram, as given in
the following Figure 2.1
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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 5, October-2021
Services for Science and Education – United Kingdom
�# �#
X X X
X
k k k-1 k-1
Y Y
Y
Y
�! �!
Diagram A Diagram B
Figure 2.1: The illustration of semi-conjugacy and conjugacy between
�!and �"
As referred to dynamical systems, the word conjugacy means the similarity between the
dynamical behavior of two mappings. Topological conjugacy between maps is a very powerful
tool in the discussion of dynamical systems. Topological conjugacy is a significant concept in
dynamical systems. It is an essential tool that can be used to make predictions toward the
behavior of a dynamical system up to comparing it with another dynamical system whose
specific properties are known [7].
Definition 2.2:
Consider a set S which is nonempty is said to be a Cantor set if the following conditions are
satisfied: (i) S is bounded and closed, (ii) S contains no intervals, (iii) Every point in S is a limit
point of S.
Definition 2.3:
The sequence space ∑( = {(�"�#�! ... ... )}|�& = {0,1,2, ... ... , � } for all �}.That is the sequence
space ∑( is all the infinite sequences of {0, 1, 2, ... ... , �}.
Definition 2.4:
Define a projection map �: ∑! → [0,1) by �(�"�# ... �() = ∑ )!
!!"#
*
(+"
Lemma 2.1:
Consider �,� ∈ ∑! and suppose �& = �& for � = 0,1, . . . . . , �. Then, �(�,�) = #
!!. Conversely, if
�(�,�) < #
!! then �& = �&, for � = 0,1, ... ... , �.
Theorem 2.1:
The sequence space ∑( is a Cantor set that means ∑( is: