Revista
de la
Universidad
del Zulia
Fundada en 1947
por el Dr. Jesús Enrique Lossada
DEPÓSITO LEGAL ZU2020000153
ISSN 0041-8811
E-ISSN 2665-0428
Ciencias
Exactas,
Naturales
y de la Salud
Año 14 N° 40
Mayo - Agosto 2023
Tercera Época
Maracaibo-Venezuela
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The Relationships Between Discrete Dynamical Systems in Topological
Spaces and their respective Hyperextensions to sets of Compact Spaces
Héctor Méndez-Gómez *
Jorge Luís Yaulema-Castañeda **
Paulina Fernanda Bolaños-Logroño ***
Fernando Ricardo Márquez-Sañay ****
ABSTRACT
Several studies have been carried out related to the analysis of the relationship with respect
to the dynamic properties of f and its hyperextension ¯f. However, the literature regarding
the analysis of the effects of individual and collective chaos on their behaviour is scarce.
Therefore, in this article several conjectures and questions are established according to the
affectation of individual chaos in an ecosystem and its chaotic behaviour within the dynamics
of this ecosystem, but as a whole. Thus, in the first instance, an introduction to the
conceptualization of topological transitivity, chaos in the Devaney sense and how they are
specified in continuous linear operators arranged in a Fréchet space (hypercyclic operators)
will be established. In addition, the different notions of chaos that can occur depending on
the relationship of the function, and its hyperextension will be described, to finally
corroborate the present chaos with greater force than Devaney’s according to the strong
periodic specification property, the same one that applies to both f and a ¯f with the purpose
of verifying the directionality in which individual and collective chaos can occur.
KEYWORDS: Function, hyperextension, dynamic system, individual chaos, collective chaos.
*Matemático, Universidad de Costa Rica, Máster Universitario Investigación Matemática, Universitat
Politècnica de Vàlencia, España. Docente en matemática de la Universidad de Costa Rica, sede regional del
Pacífico – Costa Rica. ORCID: https://orcid.org/0000-0002-0925-8310. E-mail: hector.mendez@ucr.ac.cr
** Ingeniero Electrónico, Universidad Politécnica Salesiana, Máster Universitario en Investigación
Matemática – Universidad de Valencia. Docente en la Escuela Superior Politécnica del Chimborazo – Ecuador.
ORCID: https://orcid.org/0000-0002-0646-3984. E-mail: jorge.yaulema@espoch.edu.ec
*** Ingeniera Electrónica, Escuela Superior Politécnica del Chimborazo, Magister en Sistemas de Control y
Automatización Industrial - Escuela Superior Politécnica del Chimborazo – Ecuador, Máster Universitario en
Estadística Aplicada – Universidad de Granada - España. Docente en la Escuela Superior Politécnica del
Chimborazo – Ecuador. ORCID: https://orcid.org/0000-0003-3911-0461. E-mail:
paulina.bolanos@espoch.edu.ec
**** Ingeniero Mecánico, Escuela Superior Politécnica de Chimborazo, Máster Universitario en Matemáticas
y Computación – Universidad Internacional de la Rioja - España. Docente en la Escuela Superior Politécnica
del Chimborazo – Ecuador. ORCID: https://orcid.org/0000-0001-5549-9572. E-mail:
fernando.marquez@espoch.edu.ec
Recibido: 03/02/2023 Aceptado: 29/03/2023
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Relaciones entre sistemas dinámicos discretos en espacios topológicos
y sus respectivas hiperextensiones a conjuntos de espacios compactos
RESUMEN
Se han realizado varios estudios vinculados con el análisis de la relación con respecto a las
propiedades dinámicas de f y su hiperextensión ¯f. Sin embargo, es escasa la literatura
respecto al análisis de los efectos del caos individual y colectivo sobre sus comportamientos.
Por lo tanto, en el presente artículo se establecen varias conjeturas e interrogantes de acuerdo
a la afectación del caos individual en un ecosistema y su comportamiento caótico dentro de
la dinámica de este ecosistema, pero en su conjunto. Así, se establecerá en primera instancia
una introducción a la conceptualización de la transitividad topológica, el caos en el sentido
de Devaney y cómo se especifican en operadores lineales continuos dispuestos en un espacio
de Fréchet (operadores hipercíclicos). Además, se describirán las diferentes nociones de caos
que pueden darse según la relación de la función y su hiperextensión, para finalmente
corroborar el caos presente con mayor fuerza que el de Devaney según la propiedad de
especificación periódica fuerte, la misma que aplica tanto a f como a ¯f con el propósito de
verificar la direccionalidad en la que puede ocurrir el caos individual y colectivo.
PALABRAS CLAVE: Función, hiperextensión, sistema dinámico, caos individual, caos
colectivo.
Introduction
In the analysis of a dynamical system established in a topological space of , the
development of conceptualizations such as topological transitivity and chaos will be
determined. In the case of studying the individual chaos on the behavior of orbits in the long
term, it is necessary to determine its behavior according to the changes that can be generated
when they occur in nature, which can also be collective and establish an affectation between
both (Peris, 2005).
When studying collective chaos, an analysis based on the dynamics of the function is
developed where a family of sets predominates, which in particular will be for the study of
non-empty compacts established in a topological space (Barnsley, 1993).
Thus, a question is presented according to the different notions of chaos and the
relation that it specifies regarding a continuous function with its hyperextension
. For this purpose we will define the hyperspace of non empty compact sets
established in a topological space of that will be established by . At the same time if it
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is presented in a metric space we will be able to endow a metric Hausdorff space, same
that will allow to measure the distance between the sets.
On the other hand, chaos is defined from different notions, for instance, the one
established by Devaney and its variants both total chaos and exact chaos, as well as Li-Yorke,
the ω-chaos and distributional chaos. All of them can specify the relation according to the
function and its hyperextension The purpose is established on the comparative analysis
between the individual and the collective chaos in each of these notions and vice versa.
Moreover, a stronger chaos than the one proposed by Devaney commonly known as
the strong periodic specification property is established, which when it is applied to the
function as its hyperextension, it will determine the directionality that is specified in rising
or falling as shown in the following scheme:
1. Literature review
The description of the literature related to chaos in hyperspaces requires the initial
analysis of preliminary conceptualizations established from topological dynamics with
respect to dynamical systems and their objective on the behavior that is specified in the long
term of the iterations of a certain function on the domain points and starting on a certain
topological point (Peris, 2005). Dynamics, on the other hand, establishes its purpose
according to the study of the behavior of the orbits within a system (Barnsley, 1993). This
behavior may be equivalent to the dynamical system itself as to a conjugate system, a process
called conjugacy (Román et al., 2018).
Conjugacy is related to the equivalences that can occur in dynamical systems, where
a topological transitivity is established which is expressed by the theorem that states that
when f is weakly mixing then it will also be topologically transitive (Bauer & Sigmund, 1975),
(Ulcigrai, 2021) and (Banks, 2005).
Within the dynamics of operators, hypercyclicity is presented, which according to its
universality criterion is defined by being a succession of operators within a
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Fréchet space that is separated in . If dense subsets and of are presented as an
increasing succession of naturals
and applications of ,
(whether surely discontinuous or nonlinear, which these terms will be omitted when
referring to which will be described simply as an operator). Furthermore, an operator on
a locally convex space of will be hypercyclic when the succession is universal,
specified when there exists a vector where the orbit is defined as:
being dense in . Denoting in this case a vector as a cyclic vector for
(Liao et al., 2006).
In relation with topology, convergence has been a topic of complete interest on the
study of mathematicians, which has generated several classes of spaces, being Fréchet one of
the pioneers in completely characterizing the classes of spaces by means of convergent
sequences (Martínez, 2000).
From hyperspaces the topology can be analyzed from Vietoris that specifies the dense
set of periodic points according to a dynamic system , being necessary for to
possess a chaotic condition conforming to that established by Devaney chaos where it is
stated that a dense set of periodic points will be equivalent for the function as for its
hyperextension (Delgadillo & López, 2009).
There are several reformulations of Devaney's chaos such as the total chaos, which
states that a dynamical system being in a metric space of will establish that is
totally transitive if it is evident that the iterations of for each are topologically
transitive. It is further mentioned that if were fully transitive it will be topologically
transitive so it will suffice to consider .
Conversely, Devaney's exact chaos states that a dynamical system set in a
topological space of , will determine to be topologically exact when the totality of the
nonempty open subset exists such that .
2. Materials and methods
To establish transitivity requires the use of several definitions, examples, prepositions,
lemmas and theorems of topological dynamics, operators and hyperspaces, for which the
following sections are developed.
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2.1. Topological dynamics
Definition 1.1. The dynamical system is established according to the pair where
represents the metric space and the continuous function determined by ..
In the research, it will be simply determined as a dynamical system to ,
or else . Starting from the point ,, where its iterations are defined as ,
, . Where is the identity function on .
Definition 1.2. When the dynamical system is at the point , the set of the
orbit under will be:
The study of the orbit with respect to the point under the function
establishes the succession, thus determining equivalence with the point under
the function
In a metric space is defined the of by means of the set where
it will be established the totality of the limit points present in the orbit which is precise as
a succession.
Example 1.3. When o we specify in the following formula the iterates of ,
to denote:
Where it is determined that if la will tend to 0. On the other hand
if the will tend to
Definition 1.4. The relation of two dynamical systems such and can be
presented by the continuous function . Whereby, in the dynamical systems
and , will be called a semiconjugate of as there exists a continuous function
as. If , dense range were to be presented be the diagram will present
commutativity, as presented below:
On the other hand, in the case that is presented as a homeomorphism both and
will be called conjugates.
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Definition 1.5. Upon establishing a property in a dynamical system a semiconjugate is
preserved when in the property is satisfied by making semiconjugate
of also satisfies the aforementioned property.
Therefore, a dynamical system can be defined by another system by restricting a
subset with
Definition 1.6. If and a subset of is determined or
invariant under when
In the case when is we define as a dynamical
system.
Definition 1.7. A dynamical system is set up with the function , with
since , same that will duplicate the argument of .
Definition 1.8. In the dynamical system it is considered as topologically transitive
when any of the pairs of representing nonempty open subsets of is presented ,
being , where allows the connection of trivial parts of , topological
transitivity in existence can be deduced by means of a point containing an orbit under
dense.
Proposition 1.9. The preservation of topological transitivity is presented according to
semiconjugacy.
Proposition 1.10. If f represents a continuous function presenting a dense orbit within a
metric space of where no isolated points are evident, then will be topologically transitive
(Bauer & Karl, 1975).
Lemma 1.11. There exist certain equivalent statements to the dynamical system be:
(a) If is considered topologically transitive.
(b) On a nonempty open set of let
be dense in .
Theorem 1.12. Birkhoff determines transitivity by means of the continuous function
present in a separate and complete metric space of without presence of isolated points.
By establishing the following equivalent statements:
(a) If f is considered topologically transitive.
(b) By evidencing an orbit under dense in
If one of the two statements holds the set of points in with an orbit under dense
will represent the set dense.
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Preposition 1.13. The property is established by possessing a dense orbit in preserving itself
in semiconjugacy.
Definition 1.14. Defining a metric space without evidence of assortative points
expresses as a dynamical system possessing sensitive dependence on the initial
conditions with for each y there will exist with for
, setting: where the sensitivity constant of will be represented by
the number .
All the fact previously described is known as the butterfly effect since it establishes
that very small initial differences can lead to uncontrollable consequences. Furthermore, it
specifies the stability to which a dynamic system is subject.
Definition 1.15. Considering the dynamical system
(a) If the point is set, it is considered as a fixed point of when
(b) If the point is set, it is considered as periodic point of as
. Therefore, represents the period of for .
Denoting the set of periodic points by .
A point will be periodic only if it refers to a fixed point of any of the iterations of ,
.
Preposition 1.16. Within the property of maintaining a dense set of periodic points is
specified under conjugacy.
Definition 1.17. The initial version of Devaney's chaos states that a metric space without
isolated points is denoted by where represents the dynamical system that
will settle down in a chaotic sense when the conditions described below are satisfied:
(a) has a sensitive dependence on the conditions established initially.
(b) When is considered topologically transitive.
(c) If has a dense set of points that are periodic.
Example 1.18. To establish the dependence sensitive to initial conditions we require
which is determined by . Sea
this when then is defined to possess such a dependence relative to the usual
metric .
By defining with the metric , it will be equivalent to:
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Therefore, will not have such a dependence on d. In both cases is conjugate by
the identity.
Theorem 1.19. Proposed by Banks, Brooks, Cains, Davis and Stacey specify that in a metric
space without isolated points with a dynamical system will be topologically
transitive when it has a dense set of periodic points. Hence, will have a sensitive
dependence on the initial conditions with respect to the metric that is defined in the topology
of (Ulcigrai, 2021).
Definition 1.20. Devaney chaos describes a dynamical system as chaotic if it
satisfies the following conditions:
(a) When is considered topologically transitive.
(b) If possesses a dense set of points that are periodic.
Both conditions are preserved under conjugacy.
Proposition 1.21. Devaney chaos is preserved by semiconjugacy.
Definition 1.22. The property of being mixable has greater force with respect to topological
transitivity, the same being expressed by on any of the nonempty open subsets
belonging to where there exists , determining:
Where it is observed that the function is mixable as well as topologically transitive.
Preposition 1.23. The property to be preserved mixable will depend on the semiconjugacy.
Having two metric spaces be and set by their Cartesian product
which will also be a metric space set by the metric:
let and be metrics that were defined respectively on and .
Within one of the bases of the topology states that the metric induced on the Cartesian
product will be composed of the products of the open subsets y .
Definition 1.24. The dynamical systems and will have a function
defined by:
is a continuous function whose iterations are set according to:
Similarly, products with more than two spaces or functions will be defined.
Theorem 1.25. The dynamical and shall satisfy the following statements:
(a) When possesses a dense orbit and will also possess a dense orbit.
(b) When is topologically transitive and will be as well.
(c) When is chaotic and will be chaotic too.
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(d) When are topologically transitive and one of them is mixable will
be so too.
(e) When is mixable only if are mixable as well.
Definition 1.26. will be weakly mixable when is topologically transitive.
In the case of the products of open sets generating a topological basis
will establish that the function will be weakly mixable if and only if in any of the 4-
tuple of nonempty open subsets of where there exists ,, it will be:
Proposition 1.27. The property of weakly mixable will hold according to semiconjugation.
Moreover, established dynamical systems and , in the case where
is weakly mixable and will be weakly mixable as well.
Definition 1.28. The dynamical system on any pair of sets be , we express
the return set of both and as follows:
Generally will be omitted since there is no ambiguity. In this definition will be
topologically transitive or mixable if and only if the return set in any of the pairs of nonempty
open sets belonging to is possessed.
Moreover, if and only if f for any of the 4-tuple of the non-empty open
subsets of , it will be obtained:
Lemma 1.29. The 4-set trick is established according to the dynamical system and
the nonempty open subsets of , let, by determining:
(a) As there exists a continuous function let which commutes with , then:
There will therefore exist non-empty open sets such as
, let
Let be topologically transitive, it is established that .
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(b) As is topologically transitive, it is determined:
Theorem 1.30. If represents a weakly mixable dynamical system, the
, will be times weakly mixable at (Banks, 2005).
Proposition 1.31. The dynamical system will be weakly mixable when the
nonempty open sets , are determined as:
Preposition 1.32. The dynamical system will be weakly mixable when any pair of
nonempty open sets , is determined as:
What will characterize the weakly mixable property with respect to the size terms
present in the return sets with equivalence in topological transitivity in certain
subsubsessions
Definition 1.33. The syndetic of a strictly increasing sequence of positive integers , is
defined according to:
On the other hand, the syndetic in a set is established according to the
succession of positive integers if is syndetic or in turn if its complement does not present
intervals of extremely large length.
Theorem 1.34. According to a dynamical system its equivalence is established by
the following conditions:
(a) If it is weakly mixable
(b) In the case of any pair of nonempty open sets where possesses
extremely large intervals of length.
(c) In the case of any syndetic sequence let the sequence be
topologically transitive.
2.2. Dynamics of operators
The dynamics to be used in this section is set according to the operators ,
where X will represent the Fréchet space.
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Definition 1.35. The pair represents a linear dynamical system consisting of a
separable Fréchet space of and an operator both linear and continuous. Both
terms will henceforth be omitted simply referred to as the operator.
Definition 1.36. The operator is considered hypercyclic when in whose
orbit under is dense. In this case will be a hypercyclic vector for . Hypercyclic vectors
for will be represented by .
Definition 1.37. The operator and a vector is called cyclic for when the
space evolved by the orbits is dense in , being:
The vector is called supercyclic for when its projective orbit is represented as:
Being dense in X.
In Fréchet spaces when there are no isolated points by means of Birkhoff's transitivity
theorem, it will facilitate the determination when an operator is hypercyclic.
Theorem 1.38. Birkhoff's transitivity theorem determines that an operator is considered
hypercyclic only when it is topologically transitive, where the hypercyclic vector set
occurs in a dense set . Several examples are described in (4).
Example 1.39. The Rolewicz operators, consider on , is
specified:
When we obtain for the totality of
as in , which determines that may not be considered as hypercyclic.
However, when considers to be hypercyclic. In the open and nonempty
subsets of one may obtain and in turn as follows:
Proposition 1.40. By means of Proposition 1.9 the hypercyclicity to be preserved by
semiconjugation is established.
Definition 1.41. The definition stated for chaos was described in Theorem 1.38 which
establishes a new statement of chaos but with a linear perspective. Where an operator will
be chaotic from Devaney's perspective if the following conditions are satisfied:
(a) Let be hypercyclic.
(b) When has a dense set of periodic points.
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Proposition 1.42. Considering a cyclic operator will have sensitive dependence with
respect to the initial conditions of any of the invariant metrics defined by the translations of
a topology .
Lemma 1.43. On a nonempty open set where is a Fréchet space there will exist a
nonempty open set , besides a 0-neighborhood, , which sets . When
presents a neighborhood of 0 there will exist a 0-neighborhood let .
The proof can be evidenced in Lemma 2.36.
Within the mixable property it is required that the return sets , referred to
as nonempty open sets of are presented as cofinite.
Proposition 1.44. The operator is considered to be mixable if or only if the return sets are
cofinite on any nonempty set of or 0-neighborhood .
Definition 1.46. If and ares Fréchet space then the following space is Fréchet space:
The operators y determined on Fréchet
spaces and are set to the operator which is defined by:
Proposition 1.47. The operators and , when the operator is
defined to be hypercyclic both and will be hypercyclic.
Proposition 1.48. The operators y , will be hypercyclic when at least one
of them is mixable determining that is hypercyclic or in turn if and only if and
were mixable.
In the contextualization of the investigation an operator is defined as
weakly mixable only when were hypercyclic or in turn if and only if for the non-empty
open subsets of ; will be obtained.
From the observation, therefore, a chain of implications concerning the operators is
precise:
" mixable" weakly mixable" hypercyclic."
Theorem 1.49. For hypercyclic operators inside Banach spaces there will exist several which
do not occur as weakly mixable (Liao et al., 2006).
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Lemma 1.50. being a hypercyclic operator any pair of nonempty open sets of or a 0-
neighborhood will establish both a nonempty open set and a 0-
neighborhood , let:
Theorem 1.51. Since is a hypercyclic operator and on any nonempty open set or 0-
neighborhood there exists a continuous operator commuting with , as:
Where it is established that will be weakly mixable.
Theorem 1.52. Since is a weakly mixable operator when on any pair of nonempty open sets
or 0-neighborhood W, such that:
Theorem 1.53. When represents a hypercyclic operator and a dense subset of
according to the orbit of each is bounded in a weakly mixable way.
Corollary 1.54. Among the operators the following are considered weakly mixable:
(a) Chaotic operators.
(b) Those hypercyclic operators that have a dense set of points where the orbits
converge.
(c) Hypercyclic operators that have a dense generalized kernel.
Proposition 1.55. being an operator, it follows that:
(a) will be weakly mixable only when T is as well.
(b) will be chaotic only when T is as well.
In contrast to (a) with respect to the result of the more general (b), the following
proposition is specified.
Proposition 1.55. Chaotic operators and will be chaotic operators if and only if is
as well.
2.3. Hyperspaces
This section specifies the study of a collective dynamics, which means the described
conceptualizations of topological dynamics and of operators that will be applied to subsets
of or to the dynamics that are established by means of functions evaluated on subsets
belonging to a metric space of .
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Definition 1.57. In a topological space X the corresponding hyperspace of nonempty
compact subsets of X is denoted by:
It is denoted according to the Vietoris topology which is established according
to the sets of form
Being the non empty open subsets of .
Within the hyper space of non empty compact sets of is where the fractals (8) and
(9) which are generally compact are accommodated.
Moreover, when X represents a metric space we will call as the Hausdorff
metric which is also a complete metric space, where the topology of it coincides with that of
Vietoris.
Definition 1.58. The Hausdorff metric is established when is a metric space by
endowing , where:
with
where
By defining the Hausdorff metric on neighborhoods of sets it is established that will
be an empty set of a metric space with ε-neighborhood being the set
Since and are non-empty subsets of , it will then be defined:
Both definitions have overlaps but in some cases the use of one is more accurate than
the other.
Definition 1.59. The set is considered fully bounded when in all there is a finite
subset belonging to as
. Denote as the ball of
center and radius referring to the metric .
Proposition 1.60. A metric space is considered when it is exposed to the
Hausdorff metric . This proof follows the ideas put forward in (Banks et al., 1992) and
(Grosse-Erdmann & Peris-Manguillot, 2011).
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Lemma 1.61. When and it will be said that there exists such that
.
Set y we define:
Proposition 1.62. For all as in the set will be closed. By the result and
the convergence of Cauchy sequences will provide the completeness proof of .
Theorem 1.63. Since y will therefore establish only when
.
Lemma 1.64. Application Lemma which states that it will be a Cauchy sequence when
it is in and in turn represents an increasing sequence of positive integers. For its
part being a Cauchy sequence in such that in all , hence, a Cauchy
sequence will exist when with respect to such that for its totality and
in all .
Lemma 1.65. When is a sequence present in and represents the set that groups
all points puntos in such a way there will exist a sequence where it converges
to and furthermore satisfies at all n. Since is a Caunchy sequence then
will be closed and nonempty.
To prove that it must be verified that is completely bounded, then in the
following lemma is specified a tool for that purpose which is to prove.
Lemma 1.66. When is a sequence of sets that is totally bounded in and furthermore
represents a subset of any of . Moreover if for a positive integer is presented as
then A will be fully bounded. The primary result of this section is described in
the following theorem.
Theorem 1.67. It is therefore established if is a complete metric space, will
be as well.
Theorem 1.68. The following theorem represents the completion of this section which can
also be visualized in (Furstenberg, 1967). Where it is expressed that if is considered as
a separable metric space then will be as well.
3. Results
Having established the literature review as well as the methods described previously,
the necessary approaches are established for the demonstration by means of the analysis of
the properties of transitivity and its influence on the different chaos that develop in
hyperspace.
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3.1. Transitivity and chaos in hyperspaces
In a metric space the concepts concerning topological transitivity and
Davaney's chaos to be abbreviated by DEV C were studied by means of the application of
functions defined on the hyperspace .
With the continuous function we consider the function to be called
hyperextension set in hyperspace with the totality of nonempty compact subsets of
denoted by with Vietoris topology, where is naturally generated by
, where represents the image of the nonempty compact set of
under . It can be seen that is correctly defined since is continuous. So also is
continuous for which it was observed: that the nonempty open subset U of and the
continuity of concocts that is recognized as an open subset of , if is
considered open by it is seen:
The set , is also presented as open as the will remain continuous.
With both functions it is necessary to know when it can go up or down as shown in
the following scheme:
3.1.1. Topological transitivity
Topological transitivity is established according to Definition 1.20 which mentions
Devaney's chaos, considering the following:
Theorem 2.1. As is a continuous function inside a topological space , equivalent
statements will be stated, as mentioned.
(a) The function is weakly mixable.
(b) The hyperextension of the function is weakly mixable.
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(c) The hyperextension of the function is topologically transitive.
With nonempty open sets with respect to the Vietoris-based canonical basis in ,
where is evident as specified in:
According to Theorem 1.30 applied to where there exists ,, it is determined:
Where it is considered such that is for y
. Establishing in this way the compacts . It
will be , determining that:
The hyperextension is found to be weakly mixable.
In in its hyperextension determines a weakly mixable dynamical system,
which leads it to be topologically transitive as specified in Definition 1.26.
On the other hand, in when is topologically transitive and with
Proposition 1.31 it will be proved that by setting nonempty open, it is
determined that:
The nonempty open subsets of , since is topologically transitive it is
established that and are open in , where , specifying:
Thus, we obtain a non-empty compact set that determines
Let with and where there exist
such that and , which implies:
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Proposition 2.2. A continuous function on a topological space refers that is
mixable only when is also mixable.
On arbitrary open sets based on Vietoris of as is mixable, it will be possible
to find for and , which results:
It is taken where . Hence, it is set
and , determining that it is mixable the
hyperextension .
When it occurs in the opposite direction () on nonempty open subsets of it
is set such that at some . It is therefore determined
that for any let be satisfied, proving that it is mixable
function .
To summarize, the result of:
topologically transitive topologically transitive.
It represents a false result according to that specified by Roman Flores in (Román-
Flores, 2003) where it is stated that a topologically transitive function is not so for its
hyperextension .
Theorem 2.4. and Theorem 2.5. Referring to the Hypercyclicity criterion where it is stated
that an operator within a separable Banach space of when the following
statements are equivalent:
(a) satisfies the Hypercyclicity criterion.
(b) When is topologically transitive.
Assertions that states the following:
topologically transitive topologically transitive.
topologically transitive topologically transitive.
3.1.2. Dense set of periodic points
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The dense set of periodic points is set to bound to be chaotic according to the notion
established by Devaney, for this purpose, the present section is described below:
Theorem 2.6. Given the dynamical system in a metric space of , if possesses a
dense set of periodic points its hyperextension will also have one (Ulcigrai, 2021).
Definition 2.7. Since is a dynamical system, the point will be regularly
recurrent if in the whole neighbourhood of there exists let at
Lemma 2.8. Given the dynamical system on a compact metric space of in
the case where regularly in the set of all recurrent points is dense in , it will be established
that its hyperextension will have a dense set of periodic points.
It is therefore proved that every nonempty open subset possesses a periodic
point for arranged in . According to the compactness of there will exist a nonempty
open set where determining that will be the closure of . A regularly
recurring point may be found. Subsequently a positive integer in n as .
Whereby it is established that with all limit points of arranged
in , which represents . As for and also in
which allows to find the periodic point for the arranged in .
Example 2.9. In (5) consider the Cantor space from a topological abelian group
structure to define the following sum:
Setting and to
determine that:
Hence, the property of having a dense set of periodic points for and will not be
equivalent.
The described direction is not fulfilled because it has been established previously that
being topologically transitive will not necessarily determine that is topologically
transitive. It is further held that:
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Particularly arranged in Example 2.9 since it was determined that has a set of
periodic points, which is not the case with . Which would establish that Devaney's causal
properties of equivalence between the function and its hyperextension are not always
satisfied.
Theorem 2.10. The Shauder-Tychonoff fixed point is determined when let be a nonempty,
compact t convex subset arranged in a convex space of and being continuous it
will thus be established that for any . The proof of the same is described in
(Rudin, 1991).
Theorem 2.11. being a continuous linear operator arranged in a locally convex and complete
space, the following equivalences are determined:
(a) It will be chaotic in the sense established by Devaney.
(b) It will be chaotic in the sense established by Devaney.
(c) It will be chaotic
in the sense established by Devaney.
For the demonstration in the chaos of in the sense established by
Devaney, it will be weakly mixable as stated in Corollary 1.54. It is therefore determined and
in accordance with Theorem 2.1 that will be topologically transitive. Furthermore, the
relation of Theorem 2.6 will be followed. Which determines the following:
By setting as Devaney chaos, the semiconjugacy will be established where
will be chaotic in the Devaney sense as well as
.
When will be topologically transitive . Which entails that if and are
presented as nonempty open subsets of the following sets will be considered:
In they will be nonopen nonempty. It will be topologically transitive
which
therefore determines that and as
. Established that
such that . It is thus proved that possesses a dense set of periodic points which
proves the existence of periodic for
, which states that
at some .
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Being nonempty, convex and compact in addition to establishing the Schauder-Tychonoff
fixed point theorem it is concluded that there exists the point .
3.1.3. Reformulations of Devaney Chaos
In generating reformulations on the notion of Devaney chaos, conditions will be
established that will formulate the total Devaney chaos (totDev C) and the exact one (exDev
C), obtaining:
Which indicates that totDev C and exDev C with respect to the function f will possess
the same notions of chaos and respectively with their hyperextension ¯f but not reciprocally.
3.1.3.1. Total Devaney Chaos
Definition 2.12. When is a dynamical system within a metric space it will be
set that is totally transitive if for the iterations of of each to be topologically
transitive, it will suffice to take ..
Proposition 2.13. Since is a weakly mixable dynamical system it will therefore be
fully transitive. Revising in (Ulcigrai, 2021), the following proof is established:
It is demonstrated that by establishing that and as nonempty open subsets of ,
it will be the nonempty open set be in where is considered to be weakly
mixable by existing as . By the algorithm
established by division, integers and will appear such that , with
,which will continue with . That is
therefore:
This leads to
Therefore, the following chain will be fulfilled in any dynamic system
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mixable
weakly mixable
fully transitive
topologically transitive
Lemma 2.14. When be a dynamical system chaotic in the sense of Devaney and
fully transitive one will establish as weakly mixable.
By setting and to be nonempty open subsets of it is determined that,
Therefore, proposition 1.32 will conclude the above theorem.
Since is chaotic in the sense of Devaney it is stated to be topologically transitive and
to have a dense set of periodic points. In this function by its characteristic of topological
transitivity is therefore determined the existence of and with .
Having periodic points will form a dense set in which will determine the existence of a
periodic point of be as also for any , where represents
the period of , which states,
Where let , even though it is not open in general, it can be ensured
that it possesses a nonempty interior. Then, as it is known about the existence of let
and being open, an open ball having center will be determined.
As is also open with the continuity of there will appear an open ball whose
center in is determined by . Which denotes that will have a nonempty
interior and will be fully transitive, where by applying topological transitivity and
in search of an integer be,
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By establishing that with the definition of it is obtained thus
appreciating the following:
In addition, it can be observed that,
Where it is presented that and where further with
Proposition q.32 it is proved that will be weakly mixable.
Definition 2.15. When be a dynamical system , is said to possess total chaos in
the Devaney sense when the conditions described below are satisfied:
(a) When is fully transitive.
(b) When has a dense set of periodic points.
Then Leman 2.14 may be described according to the following Theorem.
Theorem 2.16. As a dynamical system possessing total chaos in the Devaney sense,
it will be weakly mixable.
Theorem 2.17. (Related to Theorem 2.1). When is a dynamical system within a
topological space the following statements will be equivalent,
(a) As is weakly mixable.
(b) As is weakly mixable.
(c) As is fully transitive.
(d) As is topologically transitive.
It is demonstrated that results from the immediate Theorem 2.1.
with the application of Proposition 2.13 it is established that the function ¯f be
fully transitive.
be immediate by establishing that ¯f is fully transitive and by definition is
topologically transitive.
again with Theorem 2.1 will establish an immediate result.
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Furthermore in Theorem 2.17 the equivalence of the topological transitivity of as the
total transitivity of in is evidenced as a new result that determines the equivalence
in the functions set on the hyperspace of both total chaos and only Devaney chaos.
Theorem 2.19. In this sense by setting in a topological space of as a dynamical
system, it holds that possesses total chaos in the Devaney sense as its hyperextension
.
Since possesses total chaos in the Devaney sense then it will also be totally transitive
with a dense set of periodic points. If such a set of is dense in , the set of periodic points
in will be dense in as Theorem 2.6 states to it. On the other hand, according to
Theorem 2.17 which determines that continues to be totally transitive so it will have
therefore total Devaney chaos. Which will establish according to the following scheme the
sense of going down, as shown below,
3.1.4. Exact Devaney Chaos
In the present section it will be proved that when there exists dynamical system
f:XX with exact chaos in the Devaney sense (exDev C), its hyperextension ¯f will also have
it.
Definition 2.20. When be a dynamical system within a topological space , it is
determined that will be topologically exact upon the occurrence in the entire nonempty
open subset of the existence of as .. It is worth mentioning the
particular simplicity in determining that every topologically exact function is overjective.
Proposition 2.21. Since is a dynamical system inside a topological space will
be presented as topologically exact so it will also be topologically transitive.
It is demonstrated for as nonempty open subsets of that is topologically exact when
there exists as , establishing for the following,
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Showing that f is topologically transitive.
Definition 2.22. As is a dynamical system, it is determined to possess exact chaos
in the sense of Devaney on by satisfying the following conditions,
(a) As f is topologically exact.
(b) As f has a dense set of periodic points.
Theorem 2.23. As is a dynamical system, within a compact metric space ,
the following statements will be equivalent:
(a) As is topologically exact.
(b) As is topologically exact.
Which is demonstrated that by assuming to be topologically exact
where will be a nonempty open subset of . Moreover, is nonempty and open in
thus determining the existence of as . Particularly it is
defined which determines that .
In turn in its reciprocal assumes that will be topologically exact, which
must be proved for any open set where there exists , as
In a sense of compactness on for each there will exist a
nonempty open set such that and where establishes the closure of .
Since is topologically exact, it is found as for each , where
will therefore be established,
Where which proves that is topologically exact.
Theorem 2.24. Since is a dynamical system, when exhibits exact chaos in the
Devaney sense it is determined that will also exhibit it. Which will
determine the lowering power as shown in the following scheme,
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3.2. Transitivity and chaos in hyperspaces
The present section establishes an introduction to other types of chaos notions such
as Li-Yorke chaos (LUC), ω-chaos (ωC) and distributional chaos (dC), which will be
established according to comparisons resulting from the function defined within a
topological space and its hyperextension that is stated in the hyperspace of non-empty
compact subsets of , . From this perspective we will try to answer if it is possible to
obtain the following scheme
It is specified that in the development of this section we present (X,d) in a compact
metric space. Determining for this purpose that be a continuous function.
3.2.1. Li-Yorke chaos and ω-chaos
For the introduction to Li-Yorke chaos the following definitions are expressed:
Definition 3.1. When a pair of points are set to be a Li-Yorke pair for f if it is present,
(a)
(b)
Definition 3.2. When a scrambled Li-Yorke subset of is set to if in all pairs of
points other than will be a Li-Yorke pair. Where establishes the cardinality of .
Definition 3.3. being a Dynamical System it will be established that happens to
be chaotic in the Li-Yorke sense if it possesses a non-enumerable scrambled Li-Yorke set.
Example 3.4. According to (18) it is determined that will be a continuous
function having a periodic point of period 3, so that f will be chaotic in the Li-Yorke sense.
Definition 3.5. A dynamical system be , shall be stated,
(a) It shall be determined as proximal when a pair of points is expressed as
and if a pair is not proximal so it shall be said to be distal.
(b) It will be determined as asymptotic when a pair of points is expressed as
.
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Proposition 3.6. With the definitions described previously, it is determined that, a pair of
points will be a Li-Yorke pair only when they are proximal and non-asymptotic and
are denoted according to the following sets respectively.
For any is called the proximal cell of the point of a set that has all proximal points
conforming to denoted by .
Definition 3.7. According to the previous definition it is determined that a point is distal
when . Therefore, a dynamical system is said to be distal when its points
are also distal. A dynamical system is proximal when every pair of points of are also
proximal. Thus establishing the following, for all .
Definition 3.8. A dynamical system is determined to be Li-Yorke sensitive when
is present such that any point y where is evident with
and with the proximal pair in the case of some will obtain .
The same manifests itself in (Akin & Kolyada, 2003).
Theorem 3.9. When is a dynamical system and f Li-Yorke sensitive it will be
established that it possesses sensitive dependence on initial conditions. From this
perspective and furthermore for any point when the proximal cell is dense
in it will be presented that is Li-Yorke sensitive.
Theorem 3.10. As a dynamical system is weakly mixable it will be established at
all as the proximal cell representing a dense subset of .
Theorem 3.11. When is established as a dynamical system the following
statements will be equivalent:
(a) will be weakly mixable.
(b) In the case of any when the proximal cell is dense in
(c) In the case of the existence of with a proximal cell dense in
(d) will be dense in .
Definition 3.12. As a dynamical system presents a subset of with at least two
points, it is called a set if for any two distinct points the following
conditions will be satisfied:
(a) When it is numerable .
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(b)
Proposal established in (Li, 1993).
Definition 3.13. When a dynamical system is set up, it is determined that will
be upon the existence of non-numerable.
Theorem 3.14. Since a dynamical system will be it is determined that
will be chaotic in the Li-Yorke sense.
The proof in (Lampart, 2003) is carried out as the evidence that the reciprocal is not true.
Theorem 3.15. The relation of the Li-Yorke chaos and the between the continuous
function and its hyperextension as asserted in (Guirao et al., 2009). In order to support the
scheme shown at the beginning of the section. Then, being a dynamical system and
the hyperextension of in the hyperspace . Therefore, the following
statements will be satisfied,
(a) When a set is presented as revolved Li-Yorke (, respectively)
there will then exist for a scrambled Li-Yorke (, respectively) set
for with equal cardinality of .
(b) When is chaotic in the Li-Yorke sense (, respectively) it will then
also be .
For (a) it is demonstrated by means of the function, where it
is denoted that,
Being an isometry, the effect will be conjugate by according to the path . Thus
. will be obtained. Hence, .
When is a scrambled Li-Yorke subset of , it will be established that and
every pair of points that are distinct from will be a Li-Yorke pair. Therefore, for it
will be held that,
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It is also determined that,
Proving that if represents a Li-Yorke pair for , it establishes , as a Li-Yorke
pair for . So the following set is considered,
With respect to (b), the proof establishes that since has chaos in the Li-Yorke sense
it will also possess a set which will be Li-Yorke scrambled. What would be established by
(a), a continuation where possesses a scrambled Li-Yorke set with equal cardinality to ,
the latter being non-numerable will determine that the scrambled Li-Yorke set of will be
as well. Thus proving that possesses chaos in the Li-Yorke sense.
According to Theorem 3.15 it will be obtained,
3.2.2. Distributional chaos
Distributional chaos had its notions at the introductory level in (23) and was later
generalised in (Balibrea et al., 2005) and (Smítal, J & Stefánková, 2004).
Definition 3.16. Since is a compact metric space. For as for and
will be,
Which defines the upper distribution function for to be
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In addition, the lower distribution function for will be
The two functions described are non-decreasing and also
When
it will be determined that
, or in turn when will be
.
Both functions describe both upper and lower bounds depending on how many
times the distance that develop on the and trajectories are less than
over the course of iterations.
Definition 3.17. Since is a dynamical system and if there exist a pair of points
such that,
and at some it will be said that possesses distributional
chaos of type 1.
and
at some it will be said that possesses
distributional chaos of type 2.
at all , where will be a non-degenerate interval it will therefore
be established that will possess distributional chaos of type 3.
It is therefore denoted by but not reciprocally.
Theorem 3.18. When let both and be conjugate dynamical systems via
, setting and to be metric space. It is determined that possesses
distributional chaos of type 1 and type 2 respectively only if possesses the same types.
It is therefore demonstrated that since is a homeomorphism conjugate to y it
will be established that . By presenting continuity in given if there exists
be in if it is determined that . Such that,
And being , it generates,
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Therefore,
Let
and
be upper distribution functions with respect to and respectively.
Similar case is presented by the continuity present in , in any of the cases of
where there exists arbitrarily small being for which
establishes that Determining,
and being lower distribution functions with respect to and .
In the case where
according to Definition 3.1 we obtain
. On the
other hand, if according to Definition 3.2 we will have .
Establishing as a consequence that if possesses distributional chaos of type 1 then will
also possess it.
So also, when let and by Definition 3.2 is generated it
will show that as holds distributional chaos of type 2 then will also possess it.
Finally, in (Balibrea et al., 2005) it is proved that distributional chaos of type 3 is not
preserved under conjugacy.
Definition 3.19. Being a dynamical system , the subset de will be set
distributively scrambled for when , as for any pair of distinct points by
holding that,
Let
in the totality of
Let at some
The pair will be considered to be distributionally chaotic for . Moreover, it is
specified that will possess distributional chaos when there exists a distributionally
scrambled set but it is not numerable for the same. Denoting the similarity between
distributional chaos and type 1 chaos.
Theorem 3.20. Since the system is a dynamical system, it follows,
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(a) When a distributionally scrambled set is evident for there will exist a
similar set for and it will have the same cardinality as .
(b) When exhibits distributional chaos, will also have it.
According to Theorem 3.20 it establishes a proof that is corroborated by Theorem 3.15
where it is stated that the function will be an isometry.
So also with respect to Theorem 3.20 it may be lowered as expressed in the following scheme,
Asserting according to Theorems 3.20 and 3.15 that they are not true in general terms.
In the theorem it is stated the compactification of integers, where it is considered a discrete
topological space defining the function in,
3.2.3. Li-Yorke Chaos for Linear Operators
When is a continuous linear operator within a Fréchet space possessing chaos in
the Li-Yorke sense, will also possess chaos in the sense of Li-Yorke according to Theorem
3.15. With this and in accordance with (Bernardes et al, 2017) as well as the Banach-Steinhaus
theorem, the following lemma is developed.
Lemma 3.22. As is a continuous linear operator inside a Fréchet space and possesses a
Li-Yorke pair a residual subset of will be established being unbounded the for
each . In addition to the detailed with the Li-Yorke chaos criterion and (Bernardes et
al, 2015) the following Theorem will be proved.
Theorem 3.23. As is a continuous linear operator inside a Fréchet space and is defined
as , it has subsequences converging to 0. In the case that
is dense, equivalence in the following statements will be determined,
(a) When possesses Li-Yorke chaos
(b) When possesses Li-Yorke chaos
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(c) When possesses a Li-Yorke pair
Corollary 3.24. When represents a Fréchet space with successions let be the basis
.. Assuming a translation with weights to the left, it is determined,
being an operator, equivalences are established in the following statements,
When possesses Li-Yorke chaos
When possesses Li-Yorke chaos
When possesses a Li-Yorke pair.
In the demonstration, it is established the set which has a dense subspace of
with all finite support successions, following as detailed in the previous theorem.
Conforming to the classical Banach spaces and furthermore by
(Bermúdez et al., 2011) it is determined that will contain Li-Yorke chaos only when,
Thus establishing the characterization in relation to the succession of weights.
3.3. Specification properties
In the present scheme, the study is made in order to know the specification properties,
which are determined as notions with greater strength than chaos in the sense of Devaney.
Therefore, it is established as an objective to analyze the possibility of rising and falling in
the scheme shown below,
SPSP being the acronym given to the strong periodic specification property.
3.3.1. Specification properties in hyperspace
We will work with as a continuous function where the compact metric
space will be represented by .
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Definition 4.1. f:XX as a continuous function will be able to satisfy the strong periodic
specification property when in the entire an integer is possessed be any integer
, such as the points and the integers,
For con , there will therefore exist a point that will satisfy
the following conditions,
By satisfying this definition in the special case especial , it will only be
determined that satisfies the periodic specification property (PSP).
By omitting condition (b) in the periodic specification property, it is established that
there exists a strong specification property in what is abbreviated as SSP (weak specification
property-WSP).
Replacing both conditions in the strong periodic specification property by,
When , it is possible to conceptualize the recurrent
strong recurrent specification property (RSSP) and in the specific case it will be said
that satisfies the recurrent weak periodic specification property (RWSP).
Proposition 4.2. Let be a continuous function defined on as a metric space.
When satisfies SSP one will have that the set of periodic points of is dense in and
furthermore is mixable.
Proposition 4.3. When defining continuous functions and on compact
metric spaces respectively as and on , the following statements must be
satisfied,
(a) When satisfies SSP it is determined that will also satisfy SSP at any .
(b) When and satisfy SSP it is determined that will also satisfy SSP.
Theorem 4.4. In a dynamical system satisfying SSP it will be established that
also satisfies it. Allowing by this theorem to go down according to the
scheme posed at the beginning of this section as observed in (Bauer & Sigmund, 1975).
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It will be obtained therefore,
Where , which shows that satisfy the strong periodic
specification property.
Determining therefore that it will be possible to go down with the strong periodic
specification property, as shown below,
3.3.2. Example
The following example is established in relation to (Guirao et al., 2009) which starts
from a function with its hyperextension that satisfies the property of strong periodic
specification and also possesses chaos in the sense of Devaney but does not have the same
or the described property.
Established the dynamical system succession
being a metric space
compact in the totality of will be considered
as a dynamical
System product.
The following lemmas will allow us to prove the mentioned.
Lemma 4.5. Let any succession
of dynamical systems which will feature every
positive integer in the whole set of recurring points, frequently of which will be dense
in , it will therefore be established,
as a product dynamical
system which will contain a dense set of recurring points in a regular manner.
Lemma 4.6. Let any succession
of dynamical systems which will present the
property of strong periodic specification, it will be established
as a
dynamical system, a product which will have the property of strong or weak recurrent
specification.
It is proved in accordance with upon the existence of a positive integer such that,
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Being the constant determined by the periodic specification property for, when
it is and . When being integers
and
as So also, when are periodic points at each
the specification property on will be obtained. Which defines the following point,
Let be any points. With direct calculation it will be verified how satisfies the
specification properties.
Lemma 4.7. When is the continuous function with the strong recurrent (weak recurrent)
specification properties, its hyperextension will possess the strong periodic specification
property (or in turn the periodic specification property).
The demonstration is established according to the satisfaction of with respect to the
recurrent strong specification property. When and the constant are set by the above
property at . It is set , sets
Being open sets
that need to cover set by balls having diameter less
than with centers , let . Therefore, it is assumed that will be equally
presented at and . Since
is the center of the ball
. Moreover, if the point is
subject to the strong recurrent specification property at the points
as in
. According to Lemma 2.8 which represents a similar
argument, it finds a point periodic within a period and specifies that,
Considering the set
It is therefore established that represents a desired periodic point.
When and is a cyclic group with elements. will be endowed
with the discrete topology. Being a topological product space
corresponding to the nonnumerable number that can be
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presented of copies of it will result that is homeomorphic with respect to the Cantor
set. Which determines that is a compact, perfect space with a basis of both closed and
open numberable sets. Base that develops according to the cylinder sets in the following way,
Let and be an arbitrary succession of elements belonging to
according to length .
In the definition of the function by the
where
it will be established that,
In all .
Lemma 4.8. When is given for it will follow that,
(a) is a continuous function
(b) will not contain periodic points in a period equal to n.
(c) When is presented, the function will be able to satisfy the strong periodic
specification property.
(d) will be topologically exact.
In the proof of (a) it is proved that by setting , it will be the preimage in any
open neighbourhood of conforming to will be open. That is, being
we will have,
Which will be open. In case it is the disjunct decomposition will be set as,
In the case of (b) by assuming the existence of the sequence
be,
By definition of it is appreciated that in the totality of , where
clearly specifying where the following cases are
considered:
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1. Being it will present in such a way that and considering the
equality in it is appreciated determining a contraindication.
2. Being it will appear in particular having subsequently determining
a contraindication.
For (c) it is proved that as there exists in each in the case of two finite
successions of points be and elements of a succession will
be found conforming to,
It is then stated that .
In (d) it is analyzed according to the same techniques described previously, that since
is a non-empty cylindrical set, it will be defined that
.
Theorem 4.9. Since f:XX exists as a topologically exact Dynamical System, it does not
satisfy the strong periodic specification property by presenting "Per" (f) as nondense in X.
However, its hyperextension ¯f containing exact Devaney chaos will.
It is demonstrated according to when presented within a
Dynamical System stated in Lemma 4.8 where each contains exact Devaney chaos and
further satisfies the strong periodic specification property, it will therefore be established as
a Cartesian product where is topologically exact including the recurrent strong
specification property as provided in Lemma 4.6. Mentioning again Lemma 4.8 it is further
determined that possesses non-periodic points in a period equal to or greater than 2.
Furthermore, the function being topologically transitive and different in its identity function
establishes that the set will not be dense. This means that does not possess exact
chaos in the sense of Devaney and also does not satisfy the property of strong periodic
specification, on the contrary its hyperextension does possess such chaos and fulfils the
property mentioned in Lemmas 4.5 and 4.7 in addition to Theorem 2.23.
If exact Devaney chaos is present in , it will also possess Devaney chaos which,
according to Theorem 2.17, will be equivalent to almost total chaos in the Devaney sense,
which determines that,
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And in turn
Since does not have a dense set of periodic points.
Conclusions
With respect to the function and its hyperextension and in accordance with the
different properties of partial chaos type analyzed, the following is established,
(a) In topological transitivity ;
(b) Fully transitive ;
(c) Exactly topological ;
(d) Weakly mixable ;
(e) Mixable ;
(f) In the density of periodic points ;
With the various notions of chaos as the periodic point density, it can be established
that,
(a) Devaney chaos (Dev C) ;
(b) In the cases: totDev C, exDev C, LYC, ω C, dC, and SPSP ;
Which determines that collectively the different notions of chaos have no implication
with those individually. But vice versa they are consistently related, except for chaos in
Devaney's sense.
References
Akin, Ethan y Kolyada, Sergiy (2003). Li-Yorke sensitivity. 4, 2003, Nonlinearity, Vol. 16, págs.
1421-1433.
Aliprantis, Charalambos y Border, Kim (2006). Infinite Dimensional Analysis. Berlin : A
hitchhiker's guide, 2006. ISBN: 978-3-540-29587-7.
Balibrea, F, Smítal, J y Stefánková, M. (2005). The three versions of distributional chaos. 4,
2005, Amer. Math. Monthly, Vol. 99.
Banks, J, y otros (1992). On Devaney's definition of chaos. 4, 1992, American Mathematical
Society, Vol. 99, págs. 332-334.
REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 14, N° 40, 2023
Héctor Méndez-Gómez et al. // The Relationships Between Discrete Dynamical Systems … 43-83
DOI: https://doi.org/10.46925//rdluz.40.04
82
Banks, John (2005). Chaos for induced hyperspace maps. Chaos Solitons Fractals Vol. 25, no. 3,
págs. 681-685.
Barnsley, Michael (1993). Fractals everywhere. Second. Boston: Academic Press Professional,
1993. ISBN: 0-12-079061-0.
Bauer, W y Sigmund, K. (1975). Topological dynamics of transformations induced on the
space of probability measures. 1975, Monatsh. Math., Vol. 79, págs. 81-92
Bauer, Walter y Sigmund, Karl (1975). Topological dynamics of transformations induced on
the space of probability measures. 1975, Monatshefte für Mathematik, Vol. 79, págs. 81-92.
Bermúdez, A, y otros (2011). Li-Yorke and distributionally chaotic operators. 1, 2011, J. Math.
Anal. Appl., Vol. 373, págs. 83-93.
Bernardes, N, Peris, A y Rodenas, F. (2017). Set-valued chaos in linear dynamics. 4, 2017,
Integral Equations Operator Theory, Vol. 87, págs. 451-463.
Bernardes, N, y otros (2015). Li-Yorke chaos in linear dynamics. 6, 2015, Ergodic Theory Dynam.
Systems, Vol. 35, págs. 1723-1745.
De la Rosa, Manuel y Read, Charles (2009). A Hypercyclic Operator Whose Direct Sum T
T is not Hypercyclic.. 2, 2009, Journal Operator Theory, Vol. 61, págs. 369-380.
Delgadillo, Gerardo y López, Miguel (2009). Espacios de Fréchet Urysohn. No 1, 2009, Vol. 8,
págs. 29-56.
Edgar, Gerald (2008). Measure, topology, and fractal geometry. Second. New York : Springer,
2008. ISBN: 978-0-387-74748-4.
Furstenberg, Harry (1967). Disjointness in Ergodic Theory, Minimal Sets, and a Problem in
Diophantine Approximation. 1, 1967, Mathematical Systems Theory, Vol. 1, págs. 1-49.
Grosse-Erdmann, Karl y Peris-Manguillot, Alfred (2011). Linear chaos. London : Universitext
Springer, 2011. ISBN: 978-1-4471-2170-1.
Guirao, J, y otros (2009). Chaos on hyperspaces. 1-2, 2009, Nonlinear Anal, Vol. 71, págs. 1-8.
Lampart, Marek (2003). Two kinds of chaos and relations between them. 1, 2003, Acta Math.
Univ. Comenian, Vol. 339, págs. 119-127.
Li, Shihai (1993). ω-Chaos and Topological Entropy. 1, 1993, Transactions of the American
Mathematical Society, Vol. 3389, págs. 243-249.
Li, Shoumei; Ogura, Yukio y Kreinovich, Vladik (2022). Limit Theorems and Applications of
Set-Valued and Fuzzy Set-Valued Random Variables. Dordrecht : Springer-
Science+Bussiness media, B.V., 2022. Vol. 43. ISBN: 978-94-015-9932-0.
Li, Tien y Yorke, James (1975). Period three implies chaos. 10, 1975, The American
REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 14, N° 40, 2023
Héctor Méndez-Gómez et al. // The Relationships Between Discrete Dynamical Systems … 43-83
DOI: https://doi.org/10.46925//rdluz.40.04
83
Mathematical Monthly, Vol. 82, págs. 985-992.
Liao, Gongfu, Wang, Lidong y Zhang, Yucheng (2006). Transitivity, mixing and chaos for a
class of set-valued mappings. 1, 2006, SCI. China, Vol. Ser. A 49, págs. 1-8.
Martínez, Félix (2000). Universalidad, Hiperciclicidad y Caos en Espacios de Frechet.
Universidad Politécnica de Valencia. 2000. Tesis doctoral.
Peris, Alfredo (2005). Set-valued discrete chaos. 1, 2005, Chaos, Solitons & Fractals, Vol. 26,
págs. 19-23.
Román-Flores (2003), A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals
17, no 1, 99-104.
Román, Heriberto, Aguirre, Iván y Ayala, Víctor (2018). Algunas conexiones entre dinámica
individual y colectiva. 11, 2018, Interciencia, Vol. 43, págs. 744-750. ISSN: 0378-1844.
Rudin, W. (1991). Functional analysis. Second. New York : Mcgrawhill, 1991. ISBN 0-07-
100944-2 .
Rudin, W.(1991). Functional analysis. New York : Mcgrawhill Inc., 1991.
Smítal, J y Stefánková, M. (2004). Distributional chaos for triangular maps. 5, 2004, Chaos
Solitons Fractals, Vol. 21, págs. 1125-1128.
Ulcigrai, Corinna (2021) Slow chaos in surface flows. Bollettino dell'Unione Matematica Italiana,
Vol. 14, págs. 231-255.
