Invariance of distributional chaos for backward shifts

TL;DR

This paper establishes a precise condition under which the backward shift operator on K"{o}the sequence spaces admits an invariant distributionally $ ext{epsilon}$-scrambled set, advancing previous results in the area.

Contribution

It provides a necessary and sufficient condition for the existence of invariant distributionally $ ext{epsilon}$-scrambled sets for backward shifts on K"{o}the spaces, improving earlier findings.

Findings

Characterization of invariant distributionally $ ext{epsilon}$-scrambled sets

Improved conditions compared to previous work

Enhanced understanding of chaos invariance in sequence spaces

Abstract

A sufficient and necessary condition ensuring that the backward shift operator on the K\"{o}the sequence space admits an invariant distributionally $\varepsilon$-scrambled set for some $\varepsilon>0$ is obtained, improving the main results in [F. Mart\'{\i}nez-Gim\'{e}nez, P. Oprocha, A. Peris, J. Math. Anal. Appl., {\bf 351} (2009), 607--615].

Equations

$$B(x_1,x_2,x_3,\dots )=(x_2,x_3,x_4,\dots )$$

$${\lambda }_p(A):=\left\{x\in {\mathbb{K}}^{\mathbb{N}}:\Vert x{\Vert }_k:={\left(\sum _{j=1}^{\infty }|x_j{a}_{j,k}{|}^p\right)}^{1/p}<\infty ,\forall k\in \mathbb{N}\right\},$$

$${\lambda }_0(A):=\left\{x\in {\mathbb{K}}^{\mathbb{N}}:\underset{j\to \infty }{\lim }{x}_j{a}_{j,k}=0,\Vert x{\Vert }_k:=\underset{j\in \mathbb{N}}{\sup }|x_j{a}_{j,k}|,\forall k\in \mathbb{N}\right\}.$$

$$d(x,y)=\sum _{n=1}^{\infty }\frac{1}{2^n}\frac{\Vert x-y{\Vert }_n}{1+\Vert x-y{\Vert }_n}.$$

$$\forall n\in \mathbb{N},\exists m>n\text{ such that }\underset{j\in \mathbb{N}}{\sup }\left|\frac{a_{j,n}}{a_{j+1,m}}\right|<+\infty ,$$

$$(x{)}_k:=x_k,x(k):=B^k(x)=(x_{k+1},x_{k+2},x_{k+3},\dots ),$$

$$x(k,n):=(x_{k+1},x_{k+2},\dots ,x_{k+n},0,0,\dots ),$$

$$x[k,n]:=(\underset{n}{\underbrace{0,0,\dots ,0}},x_{k+n+1},x_{k+n+2},\dots ).$$

$${\Phi }_{x,y}^{(n)}(t)=\left|\{0\le i<n:d(f^i(x),f^i(y))<t\}\right|.$$

$${\Phi }_{x,y}(t)=\underset{n\to \infty }{\mathrm{liminf}}\frac{1}{n}{\Phi }_{x,y}^{(n)}(t),$$

$${\Phi }_{x,y}^{\ast }(t)=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}{\Phi }_{x,y}^{(n)}(t),$$

$$\underset{k\to \infty }{\lim }\frac{1}{{\mathrm{N}}_k}\left|\{0\le i<{\mathrm{N}}_k:d(B^i(x),B^i(y))\ge \delta \}\right|=1.$$

$${\mathrm{M}}_{k+1}-{\mathrm{M}}_k\ge 4^{{\mathrm{M}}_k},$$

$$\sum _{j={\mathrm{M}}_k}^{\infty }{\left|y_j{a}_{j,k}\right|}^p<\frac{1}{2^k},$$

$$d\left(0,y\left(j,{\mathrm{M}}_{2k+2}-{\mathrm{M}}_{2k+1}\right)\right)\ge d(0,B^j(y))-\frac{1}{2k}.$$

$${\nu }_j=\{\begin{array}{ll}k\cdot |y_j|,& {\mathrm{M}}_{4k}<j\le {\mathrm{M}}_{4k+3},k\in \mathbb{N},\\ 0,& \text{otherwise}.\end{array}$$

$$\sum _{j={\mathrm{M}}_k}^{\infty }|{\nu }_j{a}_{j,k}{|}^p$$

$$\left\{k_{2n}:n\in \mathbb{N}\right\}\subset \left\{4k:k\in \mathbb{N}\right\},k_1=1,k_{2n+1}=k_{2n}+3,$$

$$d\left(0,\nu \left[j,{\mathrm{M}}_{k_{2n}}-4^{{\mathrm{M}}_{k_{2n-1}}}\right]\right)\le \frac{1}{2n}.$$

$$\displaystyle\mathscr{C}_{n,l}^{0}=\Bigg{\{}j\in\mathbb{N}:$$

$${\mathcal{C}}_{n,l}^1=\left\{j\in \mathbb{N}:{\mathrm{M}}_{k_{{\mathrm{P}}_n^l+1}}<j\le {\mathrm{M}}_{k_{{\mathrm{P}}_n^l+1}+3}\right\}-{\mathcal{C}}_{n,l}^0.$$

$${\bar{\nu }}_j=\{\begin{array}{ll}|{\nu }_j|,& j\in {\mathcal{C}}_{n,l}^0,n,l\in \mathbb{N},\\ -|{\nu }_j|,& j\in {\mathcal{C}}_{n,l}^1,n,l\in \mathbb{N},\\ 0,& \text{otherwise},\end{array}$$

$$\mathcal{D}=\bigcup _{n=0}^{\infty }{B}^n\left(\left\{\alpha \bar{\nu }:\alpha \in (0,1)\right\}\right).$$

$$d(B^j(a),B^j(b))$$

$$d(B^j(a),B^j(b))<t,$$

$${\Phi }_{a,b}^{\ast }(t)$$

$$d(B^j(\alpha \bar{\nu }),B^j(\beta \bar{\nu }))$$

$${\Phi }_{a,b}(\epsilon )$$

$$d\left(B^{i-p}(a),B^{i-p}(b)\right)$$

$$B_{\mathrm{w}}(x_1,x_2,\dots ):=({\mathrm{w}}_2{x}_2,{\mathrm{w}}_3{x}_3,\dots ).$$

Full text

Let $\mathbb{N}=\{1,2,3,\ldots\}$ and $\mathbb{Z}^{+}=\{0,1,2,\ldots\}$. According to [9], an infinite matrix $A=(a_{j,k})_{j,k\in\mathbb{N}}$ is called a Köthe matrix if for every $j\in\mathbb{N}$ there exists some $k\in\mathbb{N}$ with $a_{j,k}>0$ and $0\leq a_{j,k}\leq a_{j,k+1}$ for all $j,k\in\mathbb{N}$.

Consider the backward shift defined by

[TABLE]

on the Köthe sequence space $\lambda_{p}(A)$ determined by a Köthe matrix $A$, where, for $1\leq p<+\infty$,

[TABLE]

and, for $p=0$,

[TABLE]

It is possible to define a complete metric on $\lambda_{p}(A)$ which is invariant by translation:

[TABLE]

The operator $B:\lambda_{p}(A)\longrightarrow\lambda_{p}(A)$ is continuous and well-defined if and only if the following condition on the matrix $A$ is satisfied:

[TABLE]

where in the case of $a_{j+1,m}=0$, one has $a_{j,n}=0$ and we consider $\frac{0}{0}$ as 1 (see [9]).

For simplicity, throughout this paper, for any $x=(x_{1},x_{2},x_{3},\ldots)\in\lambda_{p}(A)$ and any $k,n\in\mathbb{Z}^{+}$, denote

[TABLE]

[TABLE]

and

[TABLE]

The notion of distributional chaos was introduced by Schweizer and Smítal [17]. Let $f:X\longrightarrow X$ be a continuous map defined on a metric space $(X,d)$. For any $x,y\in X$, $n\in\mathbb{N}$ and $t\in\mathbb{R}$, let

[TABLE]

Define lower and upper distributional functions, $\mathbb{R}\longrightarrow[0,1]$ generated by $f$, $x$ and $y$, as follows:

[TABLE]

and

[TABLE]

respectively, where where $|A|$ denotes the cardinality of set $A$. A subset $D\subset X$ is distributionally $\varepsilon$-scrambled if for any distinct points $x,y\in D$, $\Phi_{x,y}^{*}(t)=1$ for any $t>0$ and $\Phi_{x,y}(\varepsilon)=0$. A pair satisfying the above condition is called a distributionally $\varepsilon$-chaotic pair.

During the last decades, many research works were devoted to the ‘chaotic behavior’ of the backward shift operator on the Köthe sequence space (more generally, Banach or Fréchet space) (see, e.g., [1, 3, 8, 10, 11, 12, 13, 19, 20, 21, 22]). For example, Martínez-Giménez and Peris [10] obtained some characterizations for hypercyclicity and Devaney chaos under backward shift on the Köthe sequence space. Martínez-Giménez [11] provided some sufficient conditions for the operator $f(B_{\mathrm{w}})$ to be chaotic in the sense of Devaney. Then, Bermúdez et al. [1] proved some useful equivalent conditions for Li-Yorke chaos and a few sufficient criteria for distributionally chaotic operators. Characterizations of hypercyclicity and Li-Yorke chaos for weighted shifts on more general sequence spaces were obtained in [8] and [3], respectively. By employing methods developed in [1], Wu and Zhu [20] proved that for a bounded operator defined on a Banach space, Li-Yorke chaos, Li-Yorke sensitivity, spatiotemporal chaos, and distributional chaos in a sequence are all equivalent, and they are all strictly stronger than sensitivity. Further results of [13] were extended to maximal distributional chaos for the annihilation operator of a quantum harmonic oscillator in [19, 24]. In 2009, Martínez-Giménez et al. [12] provided sufficient conditions for uniform distributional chaos under backward shift. Very recently, we [18, 21, 22, 25] provided a class of characterizations for uniform Li-Yorke chaos and a sufficient condition for maximal distributional chaos under backward shift on the Köthe sequence space. Bernardes et al. [2] characterized distributional chaos for linear operators on Fréchet spaces and obtained a sufficient condition to ensure the existence of dense uniformly distributionally irregular manifolds.

For quite a long time, operator theorists have been studying the so-called cyclic vectors in connection with the (invariant) subspace problem [4, 6, 7, 15, 16]. The invariant subspace problem, which is open to this day, asks whether every Hilbert space operator possesses an invariant closed subspace other than the two trivial ones given by $\{0\}$ and the whole space. Du [5] proved that an interval map is turbulent if and only if there is an invariant scrambled set in 2005. Later, Oprocha [14] extended this approach and proved that exactly the same characterization is valid for distributional chaos. Very recently, for the full shift $(\Sigma_{2},\sigma)$ on two symbols, we [23] constructed an invariant distributionally $\varepsilon$-scrambled set for any $0<\varepsilon<\dim(\Sigma_{2})$, in which each point is transitive but is not weakly almost periodic.

In [12], Martínez-Giménez et al. proved the following:

Combining this with [18, Theorem 3.3], Wu et al. [18] provided the following question:

Being a partial answer to Question 2, this paper shall prove that the hypothesis in Theorem 1 can ensure that $B$ admits an invariant distributionally $\varepsilon$-scrambled subset for any $0<\varepsilon<\delta$ (see Theorem 3).

[TABLE]

It is not difficult to check that there exists a subsequence $\{\mathrm{M}_{k}\}_{k\in\mathbb{N}}$ of $\left\{\mathrm{N}_{k}\right\}_{k\in\mathbb{N}}$ such that for any $k\in\mathbb{N}$,

[TABLE]

[TABLE]

and that for any $\mathrm{M}_{2k}<j\leq\mathrm{M}_{2k+1}$,

[TABLE]

Define $\nu=(\nu_{1},\nu_{2},\nu_{3},\ldots)$ by

[TABLE]

Because

[TABLE]

one has $\nu\in\lambda_{p}(A)$. Applying the method of induction, it can be verified that there exists a subsequence $\left\{\mathrm{M}_{k_{n}}\right\}_{n\in\mathbb{N}}$ of $\left\{\mathrm{M}_{k}\right\}_{k\in\mathbb{N}}$ such that for any $n\in\mathbb{N}$,

[TABLE]

and that for any $\mathrm{M}_{k_{2n-1}}<j\leq\mathrm{M}_{k_{2n-1}}+2^{M_{k_{2n-1}}}$,

[TABLE]

Arrange all odd prime numbers by the natural order ‘$<$’ and denote them as $\mathrm{P}_{1},\mathrm{P}_{2},\ldots$. For any $n,l\in\mathbb{N}$, set

[TABLE]

and

[TABLE]

Take $\bar{\nu}=(\bar{\nu}_{1},\bar{\nu}_{2},\bar{\nu}_{3},\ldots)\in\lambda_{p}(A)$ with

[TABLE]

and set

[TABLE]

Clearly, $B(\mathscr{D})\subset\mathscr{D}$ and $\mathscr{D}$ is uncountable. Given any two fixed points $a$, $b\in\mathscr{D}$ with $a\neq b$, there exist $\alpha,\beta\in(0,1)$ and $p,q\in\mathbb{Z}^{+}$ such that $a=B^{p}(\alpha\bar{\nu})$ and $b=B^{q}(\beta\bar{\nu})$. Without loss of generality, assume that $p\leq q$.

Now, we assert that $(a,b)$ is a distributionally $\varepsilon$-chaotic pair for any $0<\varepsilon<\delta$.

Firstly, for any $\mathrm{M}_{k_{2n-1}}<j\leq\mathrm{M}_{k_{2n-1}}+2^{M_{k_{2n-1}}}-q$, noting that $\mathrm{M}_{k_{2n}}-4^{\mathrm{M}_{k_{2n-1}}}\leq\mathrm{M}_{k_{2n}}-(j+q)$, and applying (6), it follows that

[TABLE]

Then, given any $t>0$, there exists some $\mathrm{N}\in\mathbb{N}$ such that for any $n\geq\mathrm{N}$ and any $\mathrm{M}_{k_{2n-1}}<j\leq\mathrm{M}_{k_{2n-1}}+2^{\mathrm{M}_{k_{2n-1}}}-q$,

[TABLE]

implying that

[TABLE]

Second, to prove $\Phi_{a,b}(\varepsilon)=0$ for any $0<\varepsilon<\delta$, we consider two cases as follows:

Case 1. $p=q$ and $\alpha\neq\beta$. Noting that for any $\mathrm{M}_{k_{\mathrm{P}_{n}+1}}<j\leq\mathrm{M}_{k_{\mathrm{P}_{n}+1}+1}$ and any $1\leq i\leq\mathrm{M}_{k_{\mathrm{P}_{n}+1}+2}-\mathrm{M}_{k_{\mathrm{P}_{n}+1}+1}$, \left|\left(B^{j}\left((\alpha-\beta)\bar{\nu}\right)\right)_{i}\right|=\big{|}\left((\alpha-\beta)\frac{k_{\mathrm{P}_{n}+1}}{4}y(j)\right)_{i}\big{|}, and applying (5), it follows that for all $n\in\mathbb{N}$ with $\left|\alpha-\beta\right|\frac{k_{\mathrm{P}_{n}+1}}{4}>1$ and any $\mathrm{M}_{k_{\mathrm{P}_{n}+1}}<j\leq\mathrm{M}_{k_{\mathrm{P}_{n}+1}+1}$,

[TABLE]

This, together with (2) and (3), implies that for any $0<\varepsilon<\delta$,

[TABLE]

Case 2. $q>p$. Fix any $n\in\mathbb{N}$ with $\alpha\frac{k_{\mathrm{P}_{n}^{q-p}+1}}{4}>1$. For any $\mathrm{M}_{k_{\mathrm{P}_{n}^{q-p}+1}}<i\leq\mathrm{M}_{k_{\mathrm{P}_{n}^{q-p}+1}+1}$ and any $i+1\leq j<i+(\mathrm{M}_{k_{\mathrm{P}_{n}^{q-p}}+3}-\mathrm{M}_{k_{\mathrm{P}_{n}^{q-p}}+2})$, noting that $\left|(\alpha\bar{\nu}-B^{q-p}(\beta\bar{\nu}))_{j}\right|\geq\left|(\alpha\bar{\nu})_{j}\right|$ (as $(\alpha\bar{\nu})_{j}$ and $(B^{q-p}(\beta\bar{\nu}))_{j}$ are of different signs), it follows that

[TABLE]

Similarly to the proof of Case 1, it can be verified that for any $0<\varepsilon<\delta$, $\Phi_{a,b}(\varepsilon)=0$.

Therefore, $\mathscr{D}$ is an invariant distributionally $\varepsilon$-scrambled subset for any $0<\varepsilon<\delta$. ∎

1. (1)

Given a sequence $\{\mathrm{w}_{i}\}_{i\geq 2}$ of strictly positive scalars, consider its associated weighted backward shift

[TABLE]

According to the discussions in [11, 12], the study of chaos under a weighted backward shift can be reduced to the unweighted case, with a suitable Köthe matrix. So, for weighted backward shift, we actually have also obtained similar result. 2. (2)

Applying Theorem 3, it is easy to verify that all examples in [12] admit an invariant $\varepsilon$-scrambled subset for some $\varepsilon>0$. 3. (3)

Combining Theorem 3 with [22, Theorem 2.1], it follows that [25, Theorem 3.1] holds trivially.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11601449), the National Nature Science Foundation of China (Key Program) (No. 51534006), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013), Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02), and Scientific Research Starting Project of Southwest Petroleum University (No. 2015QHZ029).