Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M = \Gamma \backslash N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="true">\</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> be a nilmanifold and <InlineEquation ID="IEq2"> <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 non-invertible Anosov map of class <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, with one-dimensional stable bundle. If the linear part <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> is totally non-invertible, namely the eigenvalues of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> are not algebraic units, then the existence of a conjugacy from f to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> implies that the stable Lyapunov exponent at each periodic point of f coincides with the constant one of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>. In particular, when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$r > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the conjugacy is <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> along stable leaves. Conversely, If <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$r > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> is horizontally irreducible, which means that the toral endomorphism of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$(\Gamma /(\Gamma \bigcap [N, N]))\backslash (N/[N, N])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo>⋂</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="true">\</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">/</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> induced by <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$f_$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> is irreducible, then a constant periodic stable Lyapunov exponent of f implies the existence of a conjugacy from f to <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$f_*$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>f</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>.

Stable Lyapunov Spectrum Rigidity of Nilmanifold Endomorphisms

期刊论文
发表日期：

2025年2月26日

查看全文

Ruihao Gu,
Wenchao Li

摘要

Let $M = \Gamma \backslash N$ $M = Γ \ N$ be a nilmanifold and $f: M \rightarrow M$ $f : M \to M$ be a non-invertible Anosov map of class $C^r$ $C^{r}$ , $r \ge 1$ $r \geq 1$ , with one-dimensional stable bundle. If the linear part $f_*$ $f_{*}$ is totally non-invertible, namely the eigenvalues of $f_*$ $f_{*}$ are not algebraic units, then the existence of a conjugacy from f to $f_*$ $f_{*}$ implies that the stable Lyapunov exponent at each periodic point of f coincides with the constant one of $f_*$ $f_{*}$ . In particular, when $r > 1$ $r > 1$ , the conjugacy is $C^r$ $C^{r}$ along stable leaves. Conversely, If $r > 1$ $r > 1$ and $f_*$ $f_{*}$ is horizontally irreducible, which means that the toral endomorphism of $(\Gamma /(\Gamma \bigcap [N, N]))\backslash (N/[N, N])$ $(Γ / (Γ ⋂ [N, N])) \ (N / [N, N])$ induced by $f_*$ $f_{*}$ is irreducible, then a constant periodic stable Lyapunov exponent of f implies the existence of a conjugacy from f to $f_*$ $f_{*}$ .