<p>We prove that for ergodic measures with large entropy have long unstable manifolds for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> surface diffeomorphisms. Specifically, for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist constants <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that for every ergodic measure <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> with metric entropy large than <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, the set of points with the size of unstable manifolds large than <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> has <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-measure large than <i>c</i>.</p>

Ergodic Measures with Large Entropy have Long Unstable Manifolds for $C^\infty $ Surface Diffeomorphisms

期刊论文
发表日期：

2026年3月12日

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Chiyi Luo,
Dawei Yang

摘要

We prove that for ergodic measures with large entropy have long unstable manifolds for $C^\infty $ $C^{\infty}$ surface diffeomorphisms. Specifically, for any $\alpha >0$ $α > 0$ , there exist constants $\beta >0$ $β > 0$ and $c>0$ $c > 0$ such that for every ergodic measure $\mu $ $μ$ with metric entropy large than $\alpha $ $α$ , the set of points with the size of unstable manifolds large than $\beta $ $β$ has $\mu $ $μ$ -measure large than c.

Ergodic Measures with Large Entropy have Long Unstable Manifolds for \(C^\infty \) Surface Diffeomorphisms

摘要