紧李群的斜积的绝对连续谱谱
The absolute continuous spectrum of skew products of compact Lie groups
摘要：
设$X$和$G$是紧李群，$F_1:X\to X$是$C^\infty$保测流的时一映射，$ϕ:X\to G$是连续函数，$π$是$G$的有限维不可约酉表示.证明了若$π\circϕ$沿流有Dini连续的Lie导数，且与$π\circϕ$拓扑度有关的矩阵乘法算子有非零行列式，则斜积$$ T_ϕ:X\times G\to X\times G,\quad(x,g)\mapsto\big(F_1(x),g\;ϕ(x)\big), $$在$π$的子空间中有纯绝对连续谱.这一结果为紧李群的斜积谱中存在绝对连续分量提供了一个简单而普遍的判据。作为一个例子，我们考虑$F_1$是$\T^d$和$X\times G=\T^d\times\T^{d'}$，$X\times G=\T^d\times\SU(2)$和$X\times G=\T^d\times\U(2)$的遍历翻译的情况。我们的证明依赖于最近的结果，积极的交换子方法的酉算子。
英文摘要：
Let $X$ and $G$ be compact Lie groups, $F_1:X\to X$ the time-one map of a $C^\infty$ measure-preserving flow, $ϕ:X\to G$ a continuous function and $π$ a finite-dimensional irreducible unitary representation of $G$. Then, we prove that the skew products $$ T_ϕ:X\times G\to X\times G,\quad(x,g)\mapsto\big(F_1(x),g\;ϕ(x)\big), $$ have purely absolutely continuous spectrum in the subspace associated to $π$ if $π\circϕ$ has a Dini-continuous Lie derivative along the flow and if a matrix multiplication operator related to the topological degree of $π\circϕ$ has nonzero determinant. This result provides a simple, but general, criterion for the presence of an absolutely continuous component in the spectrum of skew products of compact Lie groups. As an illustration, we consider the cases where $F_1$ is an ergodic translation on $\T^d$ and $X\times G=\T^d\times\T^{d'}$, $X\times G=\T^d\times\SU(2)$ and $X\times G=\T^d\times\U(2)$. Our proofs rely on recent results on positive commutator methods for unitary operators.
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