Let (X, d) be a compact metric space, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2091_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$f:X\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> be a continuous transformation with the specification property. In this paper, we studied the non-dense orbit set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2091_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="295" /> </InlineMediaObject> <EquationSource Format="TEX">$E(z_0):=\{x\in X:z_0\notin \overline{\{f^n(x):n\ge 0\}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>∉</mo> <mover> <mrow> <mo stretchy="false">{</mo> <msup> <mi>f</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> <mo>¯</mo> </mover> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and showed that for any non-transitive point <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2091_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$z_0\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, the set <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2091_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$E(z_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is either empty or carries the same Bowen upper and lower metric mean dimension as the whole space.

Non-dense orbit sets carry full metric mean dimension

期刊论文
发表日期：

2025年5月20日

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Jiao Yang,
Ercai Chen,
Xiaoyao Zhou

摘要

Let (X, d) be a compact metric space, $f:X\rightarrow X$ $f : X \to X$ be a continuous transformation with the specification property. In this paper, we studied the non-dense orbit set $E(z_0):=\{x\in X:z_0\notin \overline{\{f^n(x):n\ge 0\}}\}$ $E (z_{0}) : = {x \in X : z_{0} \notin \bar{{f^{n} (x) : n \geq 0}}}$ and showed that for any non-transitive point $z_0\in X$ $z_{0} \in X$ , the set $E(z_0)$ $E (z_{0})$ is either empty or carries the same Bowen upper and lower metric mean dimension as the whole space.