<p>In this paper, we introduce and study concepts of mean sensitivity via the Furstenberg family. We present two definitions: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-orbit mean sensitivity and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-diameter mean sensitivity, for a countable left amenable semigroup <i>S</i>. We demonstrate that a topological dynamical system <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\left( X, \{T_s\}_{s\in S}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>X</mi> <mo>,</mo> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mi>s</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation> with a transitive point that is not <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-mean equicontinuous is <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-orbit mean sensitive. Moreover, if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\left( X, \{T_s\}_{s \in S}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>X</mi> <mo>,</mo> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mi>s</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation> is transitive, then it is either <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-orbit mean sensitive or <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-almost mean equicontinuous. Furthermore, if <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\left( X, \{T_s\}_{s \in S}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>X</mi> <mo>,</mo> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>T</mi> <mi>s</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation> is a minimal system, then it is either <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-orbit mean sensitive or <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-mean equicontinuous. Finally, we show that the definition of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathscr{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>-orbit mean sensitivity is preserved under an open factor map.</p>

$\mathscr{F}$-mean sensitivity on amenable semigroups

期刊论文
发表日期：

2026年2月3日

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Javad Jafari,
Mohammad Akbari Tootkaboni,
Abbas Sahleh

摘要

In this paper, we introduce and study concepts of mean sensitivity via the Furstenberg family. We present two definitions: $\mathscr{F}$ $F$ -orbit mean sensitivity and $\mathscr{F}$ $F$ -diameter mean sensitivity, for a countable left amenable semigroup S. We demonstrate that a topological dynamical system $\left( X, \{T_s\}_{s\in S}\right) $ $(X, {T_{s}}_{s \in S})$ with a transitive point that is not $\mathscr{F}$ $F$ -mean equicontinuous is $\mathscr{F}$ $F$ -orbit mean sensitive. Moreover, if $\left( X, \{T_s\}_{s \in S}\right) $ $(X, {T_{s}}_{s \in S})$ is transitive, then it is either $\mathscr{F}$ $F$ -orbit mean sensitive or $\mathscr{F}$ $F$ -almost mean equicontinuous. Furthermore, if $\left( X, \{T_s\}_{s \in S}\right) $ $(X, {T_{s}}_{s \in S})$ is a minimal system, then it is either $\mathscr{F}$ $F$ -orbit mean sensitive or $\mathscr{F}$ $F$ -mean equicontinuous. Finally, we show that the definition of $\mathscr{F}$ $F$ -orbit mean sensitivity is preserved under an open factor map.

\(\mathscr{F}\)-mean sensitivity on amenable semigroups

摘要