The research of $({\rm{G}}, {\rm{w}})$-Chaos and G-Lipschitz shadowing property

1.   Introduction

Chaos and shadowing property are important concepts in dynamical systems. Many scholars have studied their dynamical properties. See ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]} for relevant results. For example, Li ^[1] proved that if the self-map $f$ is $w -$ chaotic, then the shift map $\sigma$ is $w -$ chaotic; Shah, Das and Das ^[2] proved that if a uniformly continuous self-map of a uniform locally compact Hausdorff space has topological weak specification property, then it admits a topologically distributionally scrambled set of type 3; Kostic ^[3] introduced two different notions of disjoint distributional chaos for sequences of continuous linear operators in Frechet spaces; Wang and Liu ^[4] generalized the notion of the ergodic shadowing property to the iterated function systems and proved some related theorems; Wang and Zeng ^[5] studied the relationship between average shadowing property and $\underline q -$ average shadowing property; for any $k \ne 0$ and $f \in H(X)$, Li ^[6] proposed that $f$ is chaotic if and only if ${f^k}$ is chaotic; Li ^[7] proved that a chaotic semi-flow $\theta$ on a manifold $M$ in the sense of Devaney with some assumptions is an expanding semi-flow; Li and Zhou ^[8] presented that if a continuous Lyapunov stable map $f$ from a compact metric space $X$ into itself is topologically transitive and has the asymptotic average shadowing property, then $X$ is consisting of one point; Li ^[9] proved that if the set of all periodic points of $\phi \times \theta$ is dense in $X \times Y$, then $\phi \times \theta$ is chaotic.

According to the concept of $w -$ Chaos ^[1], we introduced the definition of $(G, w) -$ Chaos. Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. We say that $S$ is an $(G, w) -$ scrambled set if for any $x, y \in S$ with $x \ne y$, the following three conditions are satisfied: (1)${w_G}(x, f) - {w_G}(y, f)$ is uncountable; (2) ${w_G}(x, f) \cap {w_G}(y, f)$ is nonempty; (3)${w_G}(x, f) \not\subset {P_G}(f)$. The map $f$ is said to be $(G, w) -$ chaotic if there exists an uncountable $(G, w) -$ scrambled set. Then we proved that if the self-map is $(G, w) -$ chaotic, the shift map $\sigma$ is $(G, w) -$ chaotic in the inverse limit space under action group which generalizes the conclusion of $w -$ Chaos given by Li ^[1].

Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. The map $f$ has $G -$ Lipschitz shadowing property if there exists positive constant $L > 0$ and ${\delta _0} > 0$ such that for any $0 < \delta < {\delta _0}$ and $(G, \delta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$ there exists a point $x \in X$ such that the sequence $\{ {x_i}\} _{i = 0}^\infty$ is $(G, L\delta) -$ shadowed by the point $x$ ^[17]. Finally, the dynamical properties of $G -$ Lipschitz shadowing property are studied under topological $G -$ conjugate and iterative systems. We derive that (1) If $f$ is topologically $G -$ conjugate to $g$, then the map $f$ has $G -$ Lipschitz shadowing property if and only if the map $g$ has $G -$ Lipschitz shadowing property. (2) For any positive integer $k \geqslant 2$, the map $f$ has the $G -$ Lipschitz shadowing property if and only if the iterative map ${f^k}$ has the $G -$ Lipschitz shadowing property. These results enrich the theory of topological $G -$ conjugate and iterative system.

Next, I will give the proof of above three conclusions in sections 2–4.

2.   (G, w)-Chaos in the inverse limit space under group action

Definition 2.1. ^[18] Let $(X, d)$ be a metric $G -$ space, $G$ be a topological group and $\varphi$ be a continuous map from $G \times X$ to $X$. The $(X, G, \varphi)orX$ is called to be a metric $G -$ space if the following conditions are satisfied:

(1) $\varphi (e, x) = x$ for all $x \in X$ where $e$ is the identity of $G$;

(2) $\varphi ({g_1}, \varphi ({g_2}, x)) = \varphi ({g_1}{g_2}, x)$ where for all $x \in X$ and all ${g_1}$, ${g_2} \in G$.

If $X$ is compact, then $X$ is also said to be a compact metric $G -$ space. For the convenience, $\varphi (g, x)$ is usually abbreviated as $gx$.

Definition 2.2. ^[19] Let $(X, d)$ be a metric space and $f$ be a continuous map from $X$ to $X$. We say that ${X_f}$ is the inverse limit spaces of $X$ if we write

Where let $\underleftarrow {\lim }(X, f)$ denoted by the inverse limit spaces ${X_f}$.

The metric $\bar d$ in ${X_f}$ is defined by

where $\bar x = ({x_0}, {x_1}, {x_2} \cdot \cdot \cdot) \in {X_f}$ and $\bar y = ({y_0}, {y_1}, {y_2} \cdot \cdot \cdot) \in {X_f}$.

The shift map $\sigma :{X_f} \to {X_f}$ is defined by

For every $i \geqslant 0$ the projection map ${\pi _i}:{X_f} \to X$ is defined by

Thus $({X_f}, \bar d)$ is compact metric space and the shift map $\sigma$ is homeomorphism.

Definition 2.3. ^[19] Let $(X, d)$ be a metric $G -$ space and $f$ be equivariant map from $X$ to $X$. Write

where ${G_i} = G, i \geqslant 0$.

The map $\varphi :\bar G \times {X_f} \to {X_f}$ is defined by

where $\bar g = (g, g, g \cdot \cdot \cdot) \in \overline G$ and $\bar x = ({x_0}, {x_1}, {x_2} \cdot \cdot \cdot) \in {X_f}$.

Then $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ is a metric $G -$ space.

Let $(X, G, d, f)$ and $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ are shown as above. The space $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ is called to be the inverse limit spaces of $(X, G, d, f)$ under group action.

Definition 2.4. ^[20] Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$.We say that the map $f$ is an equivariant map if we have $f(px) = pf(x)$ for all $x \in X$ and all $p \in G$.

Definition 2.5. ^[21] Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. The point $x$ is called to be an $G -$ periodic point if there exists positive integer $m$ and $g \in G$ such that $g{f^m}(x) = x$. Denoted by ${P_G}(f)$ the $G -$ periodic point set of $f$.

Definition 2.6. ^[4] Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. The point $y$ is said to be an $G -$ limit point of the point $x$ if there exists $\{ {n_i}\} \subset {N_ + }$ and $\{ {g_i}\} \subset G$ such that $\mathop {\lim }\limits_{i \to \infty } {g_i}{f^{{n_i}}}(x) = y$. Denoted by ${w_G}(x, f)$ the $G -$ limit point set of the point $x$.

Definition 2.7. ^[21] Let $(X, d)$ be a metric space and $f$ be a continuous map from $X$ to $X$. If $f(A) \subset A$ then we say that the set $A$ is invariant to the map $f$.

Definition 2.8. ^[1] Let $(X, d)$ be a metric space and $f$ be a continuous map from $X$ to $X$. We say that $S$ is an $w -$ scrambled set if for any $x, y \in S$ with $x \ne y$ the following three conditions are satisfied:

(1) $w(x, f) - w(y, f)$ is uncountable;

(2) $w(x, f) \cap w(y, f)$ is not empty;

(3) $w(x, f) \not\subset P(f) .$

We say that the map $f$ is $w -$ chaotic if there exists an uncountable $w -$ scrambled set.

Remark 2.9. According to the definition of $w -$ Chaos in metric space, we give the concept of $(G, w) -$ Chaos in metric $G -$ space.

Definition 2.1. ^[1] Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. We say that $S$ is an $(G, w) -$ scrambled set if for any $x, y \in S$ with $x \ne y$ the following three conditions are satisfied:

(1) ${w_G}(x, f) - {w_G}(y, f)$ is uncountable;

(2) ${w_G}(x, f) \cap {w_G}(y, f)$ is not empty;

(3) ${w_G}(x, f) \not\subset {P_G}(f) .$

We say that the map $f$ is $(G, w) -$ chaotic if there exists an uncountable $(G, w) -$ scrambled set.

Definition 2.11. ^[22] Let $(X, d)$ be a metric space and $f$ be a continuous map from $X$ to $X$. The set $A$ is said to be a minimal set if for any $x \in A$ we have $\overline {orb(x, f)} = A$.

Lemma 2.12. ^[21] Let $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ be the inverse limit space of $(X, G, d, f)$ under group action, $f$ be an equivariant homeomorphism map from $X$ to $X$ and $\bar x = ({x_0}, {x_1}, {x_2} \cdot \cdot \cdot) \in {X_f}$. Then we have

Lemma 2.13. ^[21] Let $(X, d)$ be a metric $G -$ space, $f$ be an equivariant homeomorphism map from $X$ to $X$ and $x \in X$. Then we have that ${w_G}(x, f)$ is closed and

Lemma 2.14. ^[21] Let $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ be the inverse limit space of $(X, G, d, f)$ under group action and $f$ be an equivariant homeomorphism map from $X$ to $X$. Then we have

Theorem 2.15. Let $({X_f}, \bar G, {\text{ }}\bar d, \sigma)$ be the inverse limit space of $(X, G, d, f)$ under group action and $f$ be an equivariant homeomorphism map from $X$ to $X$. If the self-map $f$ is $(G, w) -$ chaotic, then the shift map $\sigma$ is $(G, w) -$ chaotic.

Proof. Suppose that the self-map $f$ is $(G, w) -$ chaotic. Then there exists an uncountable $(G, w) -$ scrambled set $S$. Write

Thus $D'$ is an uncountable set in ${X_f}$. Next, we will show that $D'$ is an $(G, w) -$ scrambled set in ${X_f}$. Let

According to that $f$ is an homeomorphism map, the point ${x_0}$ and ${y_0}$ are two different points in $S$. By the definition of $(G, w) -$ scrambled set $S$, we have the following three conditions:

(1) ${w_G}({x_0}, f) - {w_G}({y_0}, f)$ is uncountable;

(2) ${w_G}({x_0}, f) \cap {w_G}({y_0}, f)$ is not empty;

(3) ${w_G}({x_0}, f) \not\subset {P_G}(f) .$

Firstly, we will show that ${w_G}(\bar x, \sigma) - {w_G}(\bar y, \sigma)$ is uncountable. Let

By Lemmas 2.12 and 2.13, we have that

Hence there exists $\bar s \in {w_G}(\bar x, \sigma)$ such that ${\pi _0}(\bar s) = {s_0}$. If $\bar s \in {w_G}(\bar y, \sigma)$ then there exists positive integer sequence $\{ {n_i}\} _{i = 0}^\infty$ and ${\bar g_i} = ({g_i}, {g_i}, {g_i} \cdot \cdot \cdot) \in \bar G$ such that

Thus $\mathop {\lim }\limits_{i \to \infty } {g_i}{f^{{n_i}}}({y_0}) = {s_0}$. So ${s_0} \in {w_G}({y_0}, f)$ which is absurd. Hence $\bar s \notin {w_G}(\bar y, \sigma)$. Thus, we have that

That is,

Then we can get that

According to that the set ${w_G}(x, f) - {w_G}(y, f)$ is uncountable, we can get that the set

is uncountable. Hence the set ${w_G}(\bar x, \sigma) - {w_G}(\bar y, \sigma)$ is uncountable.

Secondly, we will show that ${w_G}(\bar x, \sigma) \cap {w_G}(\bar y, \sigma)$ is not empty. By Lemma 2.12, we have that

Since ${w_G}({x_0}, f) \cap {w_G}({x_0}, f)$ is a nonempty closed invariant subset and $X$ is compact metric space, there exists a minimal set $M$ in ${w_G}({x_0}, f) \cap {w_G}({x_0}, f)$. Hence, we can get that

So, we have that

Finally, we will show ${w_G}(\bar x, \sigma) \not\subset {P_G}(\sigma)$. Suppose ${w_G}(\bar x, \sigma) \subset {P_G}(\sigma)$. By Lemmas 2.12 and 2.14, we have that

Hence, we can get that

So, we have that

That is

Thus, the assumption is absurd. So ${w_G}(\bar x, \sigma) \not\subset {P_G}(\sigma)$. Hence the set $D'$ is an uncountable $(G, w) -$ scrambled set in ${X_f}$. So, the shift map $\sigma$ is $(G, w) -$ chaotic. Thus, we complete the proof.

3.   G-Lipschitz shadowing property under topological G-conjugate

Definition 3.1. ^[16] Let $(X, d)$ be a metric space and $f$ be a continuous map from $X$ to $X$. The map $f$ is said to be a Lipschitz map if there exists a positive constant $L$ such that for all $x, y \in X$ implies

Definition 3.2. ^[17] Let $(X, d)$ be a metric $G -$ space and $f$ be a continuous map from $X$ to $X$. The map $f$ has $G -$ Lipschitz shadowing property if there exists positive constant $L$ and ${\delta _0}$ such that for any $0 < \delta < {\delta _0}$ and $(G, \delta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$ there exists a point $x \in X$ such that the sequence $\{ {x_i}\} _{i = 0}^\infty$ is $(G, L\delta) -$ shadowed by the point $x$.

Definition 3.3. ^[18] Let $(X, d)$ and $(Y, d)$ be a metric $G -$ space, $f$ be a continuous map from $X$ to $X$ and $g$ be a continuous map from $Y$ to $Y$. We say that $f$ is topological $G -$ conjugate to $g$ about $h:X \to Y$ if $h \circ f = g \circ h$ and $h$ is an equivariant homeomorphism map from $X$ to $Y$.

Theorem 3.4. Let $(X, d)$ be metric $G -$ space, $(Y, d)$ be metric $G -$ space and $f$ be topologically $G -$ conjugate to $g$ about the map $h:X \to Y$. If $h$ is a Lipschitz map with Lipschitz constant ${L_1}$ from $X$ to $Y$ and ${h^{ - 1}}$ is a Lipschitz map with Lipschitz constant ${L_2}$ from $Y$ to $X$, then the map $f$ has $G -$ Lipschitz shadowing property if and only if the map $g$ has $G -$ Lipschitz shadowing property.

Proof. Suppose that the map $f$ has the $G -$ Lipschitz shadowing property. Then there exists positive constant ${L_0} > 0$ and ${\varepsilon _0} > 0$ such that for any $0 < \delta < {\varepsilon _0}$ and $(G, \delta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$ there exists a point $x \in X$ such that the sequence ${\{ {x_i}\} _{i \geqslant 0}}$ is $(G, {L_0}\delta) -$ shadowed by the point $x$. Let

For any $0 < \eta < {\varepsilon _1}$, let $\{ {y_i}\} _{i = 0}^{ + \infty }$ be $(G, \eta) -$ pseudo orbit of $g$. Then for any $i \geqslant 0$ there exists ${t_i} \in G$ satisfying

According to that ${h^{ - 1}}$ is a Lipschitz map with Lipschitz constant ${L_2}$, we can get that

According to that $h$ is an equivalent map and $h \circ f = g \circ h$, for any $i \geqslant 0$ we have that

Thus ${h^{ - 1}}({y_i})$ is $(G, {L_2}\eta) -$ pseudo orbit of $f$. According to that $f$ has the $G -$ Lipschitz shadowing property, there exists $x \in X$ such that for any nonnegative integer $i \geqslant 0$ there exists ${p_i} \in G$ satisfying

Since $h$ is a Lipschitz map with Lipschitz constant ${L_1}$, we can obtain that

According to that $h$ is an equivalent map and $h \circ f = g \circ h$, for any $i \geqslant 0$ we have that

Hence the map $g$ has $G -$ Lipschitz shadowing property.

The method is the same as above and the proof is omitted here. Thus, we complete the proof.

4.   G-Lipschitz shadowing property of iterative map

Definition 4.1. ^[23] Let $(X, d)$ be a metric $G -$ space. The metric $d$ is said to be invariant to the topological group $G$ provided that $d(x, y) = d(gx, gy)$ for all $x, y \in X$ and $g \in G$.

Definition 4.2. ^[24] Let $G$ be topological group. $G$ is said to be commutative provided that $p \cdot g = g \cdot p$ for all $p, g \in G$.

Theorem 4.3. Let $(X, d)$ be a compact metric $G -$ space, $f:X \to X$ be an equivalent Lipschitz map with Lipschitz constant $L$ and the metric $d$ be invariant to the topological group $G$ where $G$ is commutative. Then the map $f$ has the $G -$ Lipschitz shadowing property if and only if for any positive integer $k \geqslant 2$ the iterative map ${f^k}$ has the $G -$ Lipschitz shadowing property.

Proof. Suppose that the map $f$ has the $G -$ Lipschitz shadowing property. Then there exists positive constant ${L_0} > 0$ and ${\varepsilon _0} > 0$ such that for any $0 < \varepsilon < {\varepsilon _0}$ and $(G, \varepsilon) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$ there exists a point $x \in X$ such that the sequence ${\{ {x_i}\} _{i \geqslant 0}}$ is $(G, {L_0}\varepsilon) -$ shadowed by the point $x$. Let $\{ {y_i}\} _{i = 0}^{ + \infty }$ be $(G, \varepsilon) -$ pseudo orbit of ${f^k}$ and ${x_{ki + j}} = {f^j}({y_i})$ where $i \geqslant 0$ and $0 \leqslant j \leqslant k - 1$. Thus $\{ {x_i}\} _{i = 0}^\infty$ is $(G, \varepsilon) -$ pseudo orbit of $f$. According to that $f$ has the $G -$ Lipschitz shadowing property, there exists $x \in X$ such that for any nonnegative integer $i \geqslant 0$ there exists ${g_i} \in G$ satisfying

Hence for any $i \geqslant 0$ we have that

That is,

So, the iterative map ${f^k}$ has the $G -$ Lipschitz shadowing property.

Suppose that the iterative map ${f^k}$ has the $G -$ Lipschitz shadowing property. Then there exists ${L_1} > 0$ and ${\varepsilon _1} > 0$ such that for any $0 < \delta < {\varepsilon _1}$ and any $(G, \delta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of ${f^k}$ there exists a point $z \in X$ such that the sequence ${\{ {x_i}\} _{i \geqslant 0}}$ is $(G, {L_1}\delta) -$ shadowed by the point $z$.

Case1. When $L \geqslant 1$. Write

For any $0 < \eta < \frac{{{\varepsilon _1}}}{{{L_2}}}$, let $\{ {x_i}\} _{i = 0}^\infty$ be $(G, \eta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$.Then for any $i \geqslant 0$ there exists ${t_i} \in G$ satisfying

Hence for any $i \geqslant 0$ we have that

According to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$, we can get that

Since $d$ is invariant to the topological group $G$ and $G$ is commutative, we can obtain that

Let ${y_i} = {x_{ki}}, i \geqslant 0$. Thus, for any $i \geqslant 0$ we have that

Hence $\{ {y_i}\} _{i = 0}^\infty$ is $(G, {L_2}\eta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of ${f^k}$. According to that ${f^k}$ has the $G -$ Lipschitz shadowing property, there exists $z \in X$ such that for any nonnegative integer $i \geqslant 0$ there exists ${p_i} \in G$ satisfying

That is, for any $i \geqslant 0$, we have that

By Eq (1) for any $0 \leqslant j \leqslant k - 1$ and $i \geqslant 0$ we can get that

According to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$, we can get that

Since $d$ is invariant to the topological group $G$ and $G$ is commutative, we can obtain that

By Eq (2) and according to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$ and $d$ is invariant to the topological group, we can obtain that

Hence, we have that

That is,

By Eqs (2) and (3) for any $i \geqslant 0$ there exists ${s_i} \in G$ satisfying

So, when $L \geqslant 1$, the map $f$ has the $G -$ Lipschitz shadowing property.

Case2. When $0 < L < 1$. For any $0 < \eta < \frac{{{\varepsilon _1}}}{k}$, let $\{ {x_i}\} _{i = 0}^\infty$ be $(G, \eta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of $f$. Then for any $i \geqslant 0$ there exists ${t_i} \in G$ satisfying

Hence, for any $i \geqslant 0$ we have that

According to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$, we can get that

Since $d$ is invariant to the topological group $G$ and $G$ is commutative, we can obtain that

Write ${y_i} = {x_{ki}}$ where $i \geqslant 0$. We have that

Hence $\{ {y_i}\} _{i = 0}^\infty$ is $(G, k\eta) -$ pseudo orbit ${\{ {x_i}\} _{i \geqslant 0}}$ of ${f^k}$. According to that ${f^k}$ has the $G -$ Lipschitz shadowing property, there exists $z \in X$ such that for any nonnegative integer $i \geqslant 0$ there exists ${p_i} \in G$ satisfying

That is, for any $i \geqslant 0$, we have that

By Eq (4), for any $0 \leqslant j \leqslant k - 1$ and $i \geqslant 0$ we can get that

According to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$, we can get that

Since $d$ is invariant to the topological group $G$ and $G$ is commutative, we can obtain that

By Eq (5) and according to that $f$ is an equivalent Lipschitz map with Lipschitz constant $L$ and $d$ is invariant to the topological group, we can obtain that

Thus, we have that

That is,

By Eqs (5) and (6) for any $i \geqslant 0$ there exists ${s_i} \in G$ satisfying

Hence, when $0 < L < 1$, the map $f$ has the $G -$ Lipschitz shadowing property. Thus, we complete the proof.

5.   Conclusions

Firstly, we study the dynamical properties of $(G, w) -$ Chaos in the inverse limit space under group action in the paper. We obtained that the self-map $f$ is $(G, w) -$ chaotic, the shift map $\sigma$ is $(G, w) -$ chaotic. The conclusion generalizes the corresponding results of $w -$ Chaos given in Li ^[1]. Secondly, the dynamical properties of $G -$ Lipschitz shadowing property are studied under topological $G -$ conjugate and iterative systems. The following conclusions are obtained. (1) If $f$ is topologically $G -$ conjugate to $g$, then the map $f$ has $G -$ Lipschitz shadowing property if and only if the map $g$ has $G -$ Lipschitz shadowing property. (2) For any positive integer $k \geqslant 2$, the map $f$ has the $G -$ Lipschitz shadowing property if and only if the iterative map ${f^k}$ has the $G -$ Lipschitz shadowing property. These results enrich the theory of topological $G -$ conjugate and iterative system. It provided the theoretical basis and scientific foundation for the application of various shadowing property in computational mathematics and biological mathematics.

Acknowledgments

Research was partially supported by the NSF of Guangxi Province (2020JJA110021) and construction project of Wuzhou University of China (2020B007).

Conflict of interest

The author declares that there are no conflicts of interest regarding the publication of this article.