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Expansivity and Shadowing for Group Actions of Some Abelian Groups and
Some Finitely Generated
Anwer Ahmed Saleh1, Mohamed Hbaib2
1
University of Sfax, Departement of Mathematiques, Tunisia.
Collage of Basic Education Haditha University of Anbar, Iraq, anwer.math@uoanbar.edu.iq
2
University of Sfax, Departement of Mathematiques, Tunisia, mmmhbaib@gmail.com
Abstract
For actions of finitely formed groups, we define the term "shadowing property" and examine its
core attributes in the first part of this research. The development and application of a shadowing
lemma for nilpotent group behavior follows. The topic of expansivity and shadowing for finitely
generated group actions on metric spaces is discussed in the second section. Additionally, we
demonstrate that expansivity for such an action is identical to the action having a generator. We
research the chain recurrent set for shadowing actions. We discuss the difficulties with projecting
and lifting for actions with expansivity and shadowing. In addition, we study shadowing
properties of continuous actions of the groups 𝑍𝑝and 𝑍𝑝 ∗ 𝑅𝑝.
I. Introduction
In the dynamical systems theory of the shadowing, it has now reached a suitable level of
development (see monographs [1, 2] and a summary of recent findings [3]). Any approximated
trajectory (pseudotrajectory) that is close to an exact trajectory and is sufficiently accurate is
considered to possess a dynamical system's shadowing property. Shadowing theory is essential to
the notion of structural stability. The shadowing lemma is a key finding in the theory of
shadowing [4, 5]. It claims that in a constrained neighbourhood of a hyperbolic set, a dynamical
system possesses the shadowing characteristic. The subject of the relation between both the
asymptotic characteristics of the simulation model and the simulated results emerges from the
inescapable presence of numerous faults and disturbances in the modelling of dynamical systems
(DS). This query is particularly significant (and challenging) in the context of chaotic dynamics.
This issue was initially raised by D.V. Anosov [2, 3, (1967-70)] at the level of connections

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between specific trajectories of a hyperbolic system and the corresponding pseudo-trajectories1
as a crucial step in the analysis of structural stability of diffeomorphisms. R. Bowen suggested
the use of a comparable but less natural method called "specification" in the same situation. Both
methods essentially prevent mistakes from building up throughout the modelling process: in
systems with the shadowing property, each approximation can be uniformly traced by a real
trajectory across any length of time. Naturally, in chaotic systems, even a tiny inaccuracy in the
initial position can result in a huge divergence of trajectories that increases exponentially over
time. Dynamical system research has recently become interested in the shadowing property.
This asset's motivation is straightforward. Let's say we want to use computers to examine a
system called (X, f). Computers now occasionally employ approximations, such as substituting
3.14 for, all of which have a minor error term. We are left wondering whether these mistake
words become out of hand as our system develops over time. Do we now face a situation in
which these approximations conflict with the genuine mathematical solutions?
In the first part of this work, we define the term "shadowing property" for actions of finitely
formed groups and investigate its main characteristics. Following is the creation and use of a
shadowing lemma for nilpotent group behavior. In the second section, we consider expansivity
and shadowing for finitely generated group actions on metric spaces [1]. We also show that
expansivity for such an action is the same as an action with a generator [2]. For shadowing
actions, we investigate the chain recurrent set. We talk about the challenges of lifting and
projecting for actions involving expansivity and shadowing. Additionally, we investigate the
shadowing characteristics of ongoing actions of the groups 𝑍𝑃 and 𝑍𝑃*𝑅𝑃.
Dynamical Systems
We'd want to talk about how the space is changed by functions now. We aim to be able to talk
about how a function operating on the space moves a point through the space, to be more precise
[3-5]. The study of dynamical systems is what this is.
Definition (1)
The dynamic system can be referred as the pair (X, f), where a continuous function acting on the
space is f: X → X and X is a space. One must be able to describe how a point moves across space
in order to talk about the dynamics of the space. This brings up the idea of a point's orbit.
Definition (2)
Let x ∈ X and (X, T) is a dynamical system. Consequently, under "T", the orbit of x is set to O
(x, T): = {𝑇𝑛x | n ∈ ω}.

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In the investigation of the system's long-term behaviour, orbits are particularly crucial. Orbits can
have a single point, as with fixed points, a finite number of unique points, as with periodic or
preperiodic points, or an infinite number of distinct points. In the last scenario, a point's orbit
might cross every open set in the system; in that instance, we would describe the point's orbit as
dense.
Definition (3)
If for open sets pair "U" of a non-empty, V ⊂ x, there is k > 0 such that
𝑇𝐾(U) ∩ V ≠∅, A dynamical system is (x, T) and it is topologically transitive. According to
topological transitivity, points eventually travel under iteration to any other arbitrary open set
from one arbitrary open set.
We show that topological transitivity entails the existence of a dense orbit under the additional
assumption that there are no isolated points and that the existence of a dense orbit implies
topological transitivity under the additional assumptions that the space is separable and second
category.
2. Preliminaries
Group Theory Let we consider 𝛽 is a group and let ∅ is a subgroup of 𝛽 . For 𝛼 ∈ 𝛽 , the subset
𝛼∅ := {𝛼f: f ∈ ∅} (respectively ∅𝛼:= {f𝛼 : f ∈ ∅}) of 𝛽 is so-called leftward (correspondingly
right) coset of ∅ in 𝛽 . A normal subgroup 𝛽 of is referred to as a subgroup ∅ if 𝛼∅ = ∅𝛼 for all 𝛼
∈ 𝛽 . If a normal subgroup of 𝛽 𝑖𝑠 ∅, then all left cosets set of ∅ in 𝛽 , i.e., {𝛼∅ : 𝛼 ∈ 𝛽 } is a
group that goes by the name "quotient group". We indicate it by 𝛽 /∅. The left cosets number of
∅ in 𝛽 referred to as the index of ∅ in 𝛽 as indicated by 𝑖 𝛽 (∅). If 𝛽 is finite, it is evident that 𝑖 𝛽
(∅ ) = o(𝛽) / o(∅). Here, o( 𝛽 ) indicates how a group is arranged 𝛽 . For any 𝛼 ∈ 𝛽, A subset S of
𝛽 is referred to as a set of generating. There are only so many elements in existence. s₁, s₂,…..,sₙ
∈ S 𝖴 𝑆−1 such that 𝛼 = s₁s₂ ...sₙ. Here 𝑆−1:= {𝑆−1: s ∈ S}. If a group's generating set is a finite
set, then it is claimed that the group is finitely generated. For every s ∈ S, a set of generating "S"
is called as symmetric if 𝑆−1 ∈ S. Be aware that using a
generating set to describe a group is always convenient. For a topological group "𝛽", a subset of
non-empty "∅" is described as a syndetic collection, if there occurs a set of compact "K" such
that 𝛽 = K∅. observe that, this set of "K" needs not be a subgroup of 𝛽. Furthermore, without
sacrificing generalization, hence it can be supposed that the set "K" is a set of symmetric sets.
Because if "K" is not symmetrical, all of the "𝐾−1" in "K" must be added to make it symmetric.
Keep in mind that the new set remains a set of a compact with 𝛽 = K∅, because the map taking 𝛼

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to its inverse, 𝐾−1, is continuous map on 𝛽 We discover a necessary and sufficient condition for
a subset to be a syndetic set in the following result.
Proposition 1
If "𝛽" is called a topological group. Consequently, A subset of non-empty "∅" of " 𝛽 " is a set of
syndetic if a set of a compact "K" exists such that for all 𝛼 ∈ 𝛽, K𝛼 ∩ ∅ ≠ 𝜕. Proof Assume "∅"
is a set that is syndetic. As a result, "K" is a compact symmetric set for which 𝛽 = K∅. This
suggests each t ∈ 𝛽 can also be expressed as t = kf, for some k ∈ K and f ∈ ∅. thus,
𝐾−1t = f ∈ ∅. Also, 𝐾−1t ∈ Kt. Thus, K t ∩ ∅ ≠ 𝜕. But t ∈ G is arbitrary. Therefore, for each 𝛼 ∈
∅, K𝛼 ∩ ∅ ≠ 𝜕. On the other hand, imagine there is a compact set "K" in which for each 𝛼 ∈ ∅,
K𝛼 ∩ ∅ ≠ 𝜕. We can rely on the fact that "K" is a set of symmetric without losing generality. Let
t ∈ 𝛽. Additionally, k ∈ K and f ∈ ∅ in such a way that f = kt. This suggests that there f ∈ ∅ and
𝐾−1 ∈ K such that t = 𝐾−1f. Consequently, t ∈ K∅. But t in 𝛽 is arbitrary. Then, 𝛽 = K∅.
Accordingly, there occurs a set of finite "K" as well as 𝛽 = K∅. Hence, ∅ is a set of syndetic.
Expansive Group Action Definition and Properties
The topological group " 𝛽 " and the metric space "Y" with metric "j" should be considered. An
action ϕ: 𝛽 × Y → Y is supposed to be expansive at the occurrence of a constant "c > 0" in a way
that for all y, Z ∈ Y with y
≠ Z there occurs 𝛼 ∈ 𝛽 satisfying d(ϕ𝛼(y), ϕ𝛼(z)) > c. The constant "c" is referred to as a
constant of expansive for "ϕ". Consistently, "ϕ" is expansive if for each 𝛼 ∈ 𝛽, d(ϕ𝛼(y), ϕ𝛼(z)) ≤
c, then y = Z. Let ∅ be a subgroup of group 𝛽 and ϕ: 𝛽 × Y → Y is the action of 𝛽 on Y. Suppose
ϕ⁄∅: ∅ × Y → Y be expansive, at that time obviously ϕ is expansive.
Example
Let 𝑇2 = ℝ2 / ℤ2 indicate the two-dimensional additive torus. Recall that any sustained surjective
endomorphism "f: 𝑇2 → 𝑇2" is of the form f (y) =
𝑓𝐴(y) and can calculated as: -
𝑓𝐴(y) =Ay (module 1)
where "A" is regarded as integer matrix with a non-singular "2 × 2". If A lacks any eigenvalues
of modulus 1, the invertible transformation 𝑓𝐴 is wide. Take
𝐵 = (−1 1)
0 1
−1 0
𝐶 = (1 1)

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𝛽 ≤ 𝐶, 𝐵 > 0.
Let the action of 𝛽, ϕ: 𝛽 × 𝑇2 −→ 𝑇2 be given by ϕ(A, y) = Ay. If is a subgroup produced by BC,
then the matrix "BC" eigenvalues aren't of the unity modulus. (i.e., ∅ =< BC >), so, it is evident
that ϕ|∅ is expansive. So, ϕ: 𝛽 × 𝑇2−→ 𝑇2 is expansive. However, neither ϕ|<𝐵> nor ϕ|<𝐶> is
expansive. let "∅" be a finite index subgroup of a group "𝛽", then "∅" is undoubtedly a syndetic
set according to Proposition 1. Keep in mind that neither of the subgroups nor are sets of syndetic
in this example.
Theorem 1
If (y, j) and (y, ρ) are the spaces of the two compact metric in addition, if "G" is a group.
Supposing ϕ: 𝛽 × Y −→ Y and ψ: 𝛽 × Z −→ Z are conjugal acts accompanied by conjugacy f: Y
−→ Z. If "ϕ" is expansive, then so is "ψ". Proof Allow "ϕ" to be a wide-ranging action with a
wide-ranging constant "c". For c > 0, δ > 0 as the following equation: -
j (y, z) < δ ⇒ ρ(f(y), f(z)) < c.
With an expansive constant "δ", it is simple to see that " ψ " is expansive on "z". With a metric
"j", a metric space "Y" allegedly satisfies Property P if for every ε > 0 . There occurs a compact
subset "C" of "X" such that 𝑑−1([0, ε)) 𝖴 C × C = Y × Y. so,
𝑗−1([0, ε)) = {(y, z) ∈ y × y: j (y, z) < ε}.
It goes without saying that any compact metric space respects Property "P," although this does
not always have to be the case. Consider the equation y
= (0, 1) with the conventional metric "R," the set of real numbers. Therefore, "y" is not compact.
The following equation satisfies for every ε> 0.
𝑗−1((0, ε)) 𝖴 (ε, 1 – ε) × (ε, 1 – ε)
= (0, 1) × (0, 1).
Theorem 2
Let "Y" be a space of a compact metric and "G" be a group of a finitely generated with a set of
generating "S". Assume ϕ: 𝛽 × X → X is the action of expansive. If "Q" is not a constant of
expansive for " ϕ " if it is the least upper bound of the set "F" of all constants of expansive for " ϕ
". Proof that for every m ∈ N, Q + 1/ 𝑚 is not a constant of expansive "ϕ" and consequently,
𝑦′𝑚= 𝑧′𝑚 in a way that for wholly 𝛼 ∈ 𝛽
j (ϕ 𝛼 (𝑦′𝑚), ϕ 𝛼 (𝑧′𝑚)) < Q + 1/m.
Let δ > 0 be a constant of expansive for "ϕ". For each m, using expansivity of ϕ, there is 𝛼𝑚 ∈ 𝛽
such that
j (ϕ 𝛼𝑚 (𝑦′𝑚), ϕ 𝛼𝑚 (𝑧′𝑚)) > δ.

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Set 𝑦𝑚= ϕ𝛼𝑚 (𝑦′ 𝑚) and 𝑧𝑚 = ϕ𝛼𝑚 (𝑧′𝑚). We presume that the sequence {𝑦𝑚} and {𝑧𝑚}
converge to say, y and z correspondingly. Observe that y ≠ z. Further, for 𝛼 ∈ 𝛽 and 𝖺 > 0, select
p, q ∈ N and η > 0 such that 1/p < 𝖺/3,
j (y, 𝑦𝛼𝑞) < η ⇒ j (ϕ𝛼 (a), ϕ𝛼 (b)) < 𝖺/3
Therefore, j (ϕ𝛼 (y), ϕ𝛼 (z)) ≤ j (ϕ𝛼 (y), ϕ𝛼 (𝑧′𝑞)) + j (ϕ𝛼 (𝑧′𝑞)), ϕ𝛼 (𝑦′𝑞)
+ j (ϕ𝛼 (𝑦′𝑞), ϕ𝛼 (z)) ≤𝖺+ Q. The disparity mentioned above suggests that their y ≠ z in Y such
that for any 𝛼 ∈ 𝛽, j (ϕ𝛼 (y), ϕ𝛼 (z)) ≤ Q and therefore Q is not expansive constant.
Groups with finite generation
In this section, there are many definitions are presented, these definitions are: -
Definition (4): - A class nilpotent group 1 is any abelian group. If a group Q possesses the lower
central series of length L, it is said to be nilpotent of class L:
Q = 𝑄1 - . . . - 𝑄𝐿+1 = e, where 𝑄𝐼+1 = [𝑄𝐼, Q], 𝑄𝐿≠e (As usual, Qi 𝖺 Qi+1 means that Qi+1 is a
normal subgroup of Qi).
Definition (5): - If there is a subnormal series, a group is referred to as solvable or soluble (of not
necessarily finitely generated groups):
BS (m, L) =< a, b | 𝑏𝑎𝑚 = 𝑎𝐿 b >, m, L ∈ Z,
(As usual, Qi 𝖺 Qi+1 means that Qi+1 is a normal subgroup of Qi).
Theorem 3
Let "Q" be a free group with at least two generators that is finitely generated. Let "Φ" be a
consistent, unbroken action from "Q" on a non- discrete metric space "Ω".
a) The Φ does not have shadowing, if for some q ∈ Q the map 𝑓𝑞 is expansive.
b) For some q ∈ Q, q ≠ e, If the map "𝑓𝑞 " lacks shadowing, then " Φ " lacks shadowing as
well.
The aim of the following section is to show that A compact connected metric space's distal
homeomorphisms do not all possess the pseudo- orbit tracing characteristic.
A self-homeomorphism of Z will be used throughout this section as well as a compact metric
space with distance function d and self.
Theorem 4
Assume that X is connected. If (Z, ψ) is distal, then (Z, ψ) has not P.O.T.P.
Lemma 1. (Z, ψ) has P.O.T.P for every ε > 0 and every 𝑍𝑂 𝜖 Ω(ψ) there is a point X 𝜖 Z and an
integer k =k (𝑍𝑂, ε) > 0 such that 𝑂ψ𝐾(X) C 𝑈𝑒(𝑍0).

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Proof. Since 𝑍𝑂 𝜖 Ω(ψ), for δ > 0 with δ < ε there are a point z 𝜖 Z and the integer k > 0 such that
z and ψ𝑘(z) belong to Uδ/2(𝑍𝑂). Now, set 𝐾nk+i = ψ𝑘(z) at 0 ≤ i < k. Obviously, {𝑍𝑖} = {…., z,
ψ (z)…, ψ𝑘−1(z),}.
Hence, we can result that a point x 𝜖 Z in a way that d(ψ𝑖(X),𝑍𝑖)< ε. In particular, d(ψ𝑛𝑘(X), 𝑍𝑛𝑘)
< ε and hence d(ψ𝑛𝑘(X), Z) < ε. Therefore, we have 𝑂ψ𝐾(X) C 𝑈𝑒(𝑍0).
References
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[3] H. Hattab, Group of homeomorphisms of a metric space with finite orbits, JP.Geo.Top.
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[4] A. V. Malyutin, on groups acting on dendrons, (Russian) Zap. Nauchn. Sem. S.-Peterburg.
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[5] H. Marzougui, I. Naghmouchi, Minimal sets for group actions on dendrites, Proc. Amer.
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