论文标题
Zeta函数和海森堡周期的拓句
Zeta functions and topology of Heisenberg cycles for linear ergodic flows
论文作者
论文摘要
将dirac-schrödinger操作员沿流动的轨道放置在紧凑的歧管\（m \）上，定义了\（\ r \） - 在平滑函数（m \）上平滑函数的代数上的ecivariant光谱三重。我们研究了这些三元组的某些特性，尤其是它们的Zeta函数，它们具有\（fh^{ - s}）\）\（f \）\（f \）对\（\ r \）的限制在\（\ r \）上\（m \）和\（m \）和\（m \）和\（h = - \ frac {h = - \ frac {振荡器。这些ZETA函数的Meromormormorphic延续属性和POL结构与动力学中的Ergodic时间平均值有关。在圆圈的周期性流动的情况下，该结构重现了Lesch和Moscovici的“ Heisenberg Cycles”，在该情况下，它在非理性旋转代数\（A_ \ H \）中产生光谱三重。 我们加强了这些作者的结果，表明zeta函数\（\ trace（ah^{ - s}）\）对c*-algebra \（a_ \ h \）的任何元素\（a \）的任何元素\（a \）扩展。该构造的另一个变体会产生\（a_ \ h \ otimes a_ {1/\ h} \）的光谱周期，如果\（\ h \）满足双磷脂条件，则在合适的亚级别上具有频谱三重。该循环的类别从确定KK二元性的意义上定义了一个基本类。我们采用Connes和Moscovici的局部指数定理，以详细列出Connes的索引定理，以在线上的某些类别的差分运算符，并在基本类别引起的K理论上计算相交形式。
Placing a Dirac-Schrödinger operator along the orbit of a flow on a compact manifold \(M\) defines an \(\R\)-equivariant spectral triple over the algebra of smooth functions on \(M\). We study some of the properties of these triples, especially their zeta functions, which have the form \(\trace (fH^{-s})\) with \(f\) the restriction to \(\R\) of a function on \(M\) and \(H = -\frac{\partial^2}{\partial x^2} + x^2\) the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions is related to ergodic time averages in dynamics. The construction reproduces the `Heisenberg cycles' of Lesch and Moscovici, in the case of the periodic flow on the circle, where it produces a spectral triple over the smooth irrational torus in the irrational rotation algebra \(A_\h\). We strengthen a result of these authors, showing that the zeta function \(\trace (aH^{-s})\) extend mermomorphically for any element \(a\) of the C*-algebra \(A_\h\). Another variant of the construction produces a spectral cycle for \(A_\h\otimes A_{1/\h}\) and a spectral triple over a suitable subalgebra with the meromorphic continuation property if \(\h\) satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality. We employ the Local Index Theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class.
