One of the purposes of this article is to study the structure of periodic shadowable measures of a homeomorphism on a compact metric space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X$</EquationSource> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> </InlineEquation>, in the space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {M}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of Borel probability measures of X equipped with the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text {weak}^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>weak</mtext> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> topology. We prove that the set of periodic shadowable measures is an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_{\sigma \delta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mrow> <mi>σ</mi> <mi>δ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> subset of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {M}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the set of almost periodic shadowable measures is a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G_{\delta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation> subset of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {M}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We further prove that the almost periodic shadowable measures are <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\text {weak}^{}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>weak</mtext> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> approximated by the periodic shadowable ones. Next, we prove that the set of periodic shadowable measures is dense in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {M}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if the set of periodic shadowable points is dense in X. We construct an example of an almost periodic shadowable measure that is not periodic shadowable. The other purpose is to establish the relationship between periodic shadowable measures and shadowable measures. Further, we introduce the notion of mean asymptotic expansivity and prove that shadowable measures are periodic shadowable for such homeomorphisms.

Periodic Shadowable Measures and Mean Asymptotic Expansivity

期刊论文
发表日期：

2026年3月9日

查看全文

Pramod Kumar Das,
Priyabrata Bag

摘要

One of the purposes of this article is to study the structure of periodic shadowable measures of a homeomorphism on a compact metric space $X$ $X$ , in the space $\mathcal {M}(X)$ $M (X)$ of Borel probability measures of X equipped with the $\text {weak}^*$ ${weak}^{*}$ topology. We prove that the set of periodic shadowable measures is an $F_{\sigma \delta }$ $F_{σ δ}$ subset of $\mathcal {M}(X)$ $M (X)$ and the set of almost periodic shadowable measures is a $G_{\delta }$ $G_{δ}$ subset of $\mathcal {M}(X)$ $M (X)$ . We further prove that the almost periodic shadowable measures are $\text {weak}^{*}$ ${weak}^{*}$ approximated by the periodic shadowable ones. Next, we prove that the set of periodic shadowable measures is dense in $\mathcal {M}(X)$ $M (X)$ if and only if the set of periodic shadowable points is dense in X. We construct an example of an almost periodic shadowable measure that is not periodic shadowable. The other purpose is to establish the relationship between periodic shadowable measures and shadowable measures. Further, we introduce the notion of mean asymptotic expansivity and prove that shadowable measures are periodic shadowable for such homeomorphisms.