论文标题
Anosov差异性的联合整合性和光谱刚性
Joint integrability and spectral rigidity for Anosov diffeomorphisms
论文作者
论文摘要
令$ f \ colon \ mathbb {t}^d \ to \ mathbb {t}^d $为Anosov diffemorthism，其线性化$ a \ in {\ rm gl}（d，d，\ mathbb {z}）$是不可修复的。假设$ f $也是绝对部分夸张的，其中一个弱稳定的子围栏被认为是中心子捆绑物。我们表明，如果强稳定且不稳定的子划线共同整合，那么$ f $是动态连贯的，并且所有叶子在连接性下与线性化$ a $ a $ a $的线性叶面相对匹配。此外，$ f $承认，在弱稳定的子捆绑中，最优秀的主导地位，尺寸与$ a $相匹配的尺寸，并且沿所有这些子划分具有光谱刚度。 在维度4中，我们还能够获得类似的结果，该结果允许将弱稳定和不稳定的子捆绑成中心的子捆绑包，并假定强稳定和不稳定的子捆绑的联合整合性。作为一个应用程序，我们表明，对于{\ rm diff}^2_Ω（\ mathbb {t}^4）$的每一个符合性差异$ f \ in {\ rm diff}^2_Ω $ f $在且仅当$ f $平稳地结合到$ a $时，共同集成。
Let $f\colon\mathbb{T}^d\to\mathbb{T}^d$ be an Anosov diffeomorphism whose linearization $A\in{\rm GL}(d,\mathbb{Z})$ is irreducible. Assume that $f$ is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable and unstable subbundles are jointly integrable, then $f$ is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization $A$. Moreover, $f$ admits the finest dominated splitting in weak stable subbundle with dimensions matching those for $A$, and it has spectral rigidity along all these subbundles. In dimension 4 we are also able to obtain a similar result which allows to group the weak stable and unstable subbundles into a center subbundle and assumes joint integrability of strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism $f\in{\rm Diff}^2_ω(\mathbb{T}^4)$ which is $C^1$-close to an irreducible non-conformal automorphism $A\in{\rm Sp}(4,\mathbb{Z})$, the extremal subbundles of $f$ are jointly integrable if and only if $f$ is smoothly conjugate to $A$.
