<p>In this paper, we investigate the distributional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-chaos of the functional envelope system <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(S(\Sigma ), F_T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>F</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> induced by a nontrivial weakly mixing dynamical system <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\Sigma , T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with the shadowing property, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is an arbitrary closed subset of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A^{\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mi mathvariant="double-struck">N</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$T:\Sigma \rightarrow \Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Σ</mi> </mrow> </math></EquationSource> </InlineEquation> is a continuous map. We prove that there exists a dense Mycielski set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$K \subset S(\Sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that any <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> pairwise distinct points in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$K$</EquationSource> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> </InlineEquation> form a distributional <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-scrambled tuple, where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$d_n>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, it follows that <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$(S(\Sigma ), F_T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>F</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a distributional <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq16"> <EquationSource Format="TEX">$d_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-chaotic system with a dense Mycielski scrambled set.</p>

Distributional Chaos of Functional Envelopes on Cantor Spaces

期刊论文
发表日期：

2026年8月3日

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Huajun Gong,
Sihui Zhang,
Xiaokai Zhou

摘要

In this paper, we investigate the distributional $n$ $n$ -chaos of the functional envelope system $(S(\Sigma ), F_T)$ $(S (Σ), F_{T})$ induced by a nontrivial weakly mixing dynamical system $(\Sigma , T)$ $(Σ, T)$ with the shadowing property, where $\Sigma $ $Σ$ is an arbitrary closed subset of $A^{\mathbb {N}}$ $A^{N}$ and $T:\Sigma \rightarrow \Sigma $ $T : Σ \to Σ$ is a continuous map. We prove that there exists a dense Mycielski set $K \subset S(\Sigma )$ $K \subset S (Σ)$ such that any $n$ $n$ pairwise distinct points in $K$ $K$ form a distributional $n$ $n$ - $d_n$ $d_{n}$ -scrambled tuple, where $n \ge 2$ $n \geq 2$ and $d_n>0$ $d_{n} > 0$ . In particular, it follows that $(S(\Sigma ), F_T)$ $(S (Σ), F_{T})$ is a distributional $n$ $n$ - $d_n$ $d_{n}$ -chaotic system with a dense Mycielski scrambled set.