Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:M\rightarrow M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> be a homeomorphism on a closed manifold M. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> be a universal covering of M and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\widetilde{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> be a lifting of f to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>. We prove that topological stability for f on M induces relative topological stability for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\widetilde{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>. Conversely, we also prove that the relative topological stability for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\widetilde{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> with a leafwise condition induces topological stability for f. Finally, we construct a homeomorphism on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> which is relatively N-expansive, but is not relatively <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(N-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-expansive.

Characterizations of topological stability on universal coverings

期刊论文
发表日期：

2026年5月23日

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Hahng-Yun Chu,
Se-Hyun Ku,
Sang Hong Van Nguyen

摘要

Let $f:M\rightarrow M$ $f : M \to M$ be a homeomorphism on a closed manifold M. Let $\widetilde{M}$ $\tilde{M}$ be a universal covering of M and $\widetilde{f}$ $\tilde{f}$ be a lifting of f to $\widetilde{M}$ $\tilde{M}$ . We prove that topological stability for f on M induces relative topological stability for $\widetilde{f}$ $\tilde{f}$ on $\widetilde{M}$ $\tilde{M}$ . Conversely, we also prove that the relative topological stability for $\widetilde{f}$ $\tilde{f}$ with a leafwise condition induces topological stability for f. Finally, we construct a homeomorphism on $\widetilde{M}$ $\tilde{M}$ which is relatively N-expansive, but is not relatively $(N-1)$ $(N - 1)$ -expansive.