DEGREE, MIXING, AND ABSOLUTELY CONTINUOUS SPECTRUM OF COCYCLES WITH VALUES IN COMPACT LIE GROUPS
DEGREE, MIXING, AND ABSOLUTELY CONTINUOUS SPECTRUM OF COCYCLES WITH VALUES IN COMPACT LIE GROUPS
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紧李群中具有值的环的度、混合和绝对连续谱
DOI：
10.17654/ds030040135
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发表时间：
2016
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影响因子：
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通讯作者：
R. T. Aldecoa
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作者：
R. T. Aldecoa
我们考虑斜积 $$T_\phi:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!\phi(x)),$$，其中 $X$ 是具有概率测度的紧致流形，$G$ 是具有李代数 $\frak g$ 的紧致李群，$F_1:X\to X$ 是保测流的时间一映射，并且$\phi\in C^1(X,G)$ 是一个余循环。然后，我们将 $\phi$ 的度定义为合适的函数 $P_\phi M_\phi:X\to\frak g$，我们证明它在李群同态和 $C^1$-上同调关系下以自然的方式变换，并解释了它如何推广先前的余循环度的定义。对于$G$的每个有限维不可约表示$\pi$，以及$\pi(G)$的李代数$\frak g_\pi$，我们以类似的方式将$\pi\circ\phi$的度定义为合适的函数$P_{\pi\circ\phi}M_{\pi\circ\phi}:X\to\frak g_\pi$。如果 $F_1$ 是唯一遍历的并且函数 $\pi\circ\phi$ 对角线，或者如果 $T_\phi$ 是唯一遍历的，则 $\phi$ 的次数减少为 $\frak g$ 中的常数，由 $X$ 上的积分给出。作为副产品，如果 $G$ 是连通半单紧李群，则不存在唯一非零度的遍历偏积 $T_\phi$。
接下来，我们证明 $T_\phi$ 混合在 $P_{\pi\circ\phi}M_{\pi\circ\phi}$ 内核的正交补码中，并且在一些额外的假设下，我们表明如果 $(iP_{\pi\circ\phi}M_{\pi\circ\phi})^2$ 严格为正，则 $U_\phi$ 在该正交补码中具有纯粹绝对连续的谱。将每个 $\pi$ 的这些结果相加，即可获得混合的全局结果和 $T_\phi$ 的绝对连续谱。作为一个应用，我们提出了四种明确的情况：当 $G$ 是一个环面时，$G=SU(2)$、$G=SO(3,\mathbb R)$ 和 $G=U(2)$。在每种情况下，我们获得的结果都是新的，或者概括了以前的结果。
我们的证明依赖于酉算子的正换向器方法的新结果。
We consider skew products $$T_\phi:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!\phi(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\frak g$, $F_1:X\to X$ the time-one map of a measure-preserving flow, and $\phi\in C^1(X,G)$ a cocycle. Then, we define the degree of $\phi$ as a suitable function $P_\phi M_\phi:X\to\frak g$, we show that it transforms in a natural way under Lie group homomorphisms and under the relation of $C^1$-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irreducible representation $\pi$ of $G$, and $\frak g_\pi$ the Lie algebra of $\pi(G)$, we define in an analogous way the degree of $\pi\circ\phi$ as a suitable function $P_{\pi\circ\phi}M_{\pi\circ\phi}:X\to\frak g_\pi$. If $F_1$ is uniquely ergodic and the functions $\pi\circ\phi$ diagonal, or if $T_\phi$ is uniquely ergodic, then the degree of $\phi$ reduces to a constant in $\frak g$ given by an integral over $X$. As a by-product, we obtain that there is no uniquely ergodic skew product $T_\phi$ with nonzero degree if $G$ is a connected semisimple compact Lie group.
Next, we show that $T_\phi$ is mixing in the orthocomplement of the kernel of $P_{\pi\circ\phi}M_{\pi\circ\phi}$, and under some additional assumptions we show that $U_\phi$ has purely absolutely continuous spectrum in that orthocomplement if $(iP_{\pi\circ\phi}M_{\pi\circ\phi})^2$ is strictly positive. Summing up these results for each $\pi$, one obtains a global result for the mixing and the absolutely continuous spectrum of $T_\phi$. As an application, we present four explicit cases: when $G$ is a torus, $G=SU(2)$, $G=SO(3,\mathbb R)$, and $G=U(2)$. In each case, the results we obtain are new, or generalise previous results.
Our proofs rely on new results on positive commutator methods for unitary operators.