线性动力系统的常返性：一种基于不变测度的方法
Recurrence properties for linear dynamical systems: An approach via invariant measures
摘要：
我们从遍历论的观点研究线性动力系统的不同逐点递归概念。我们证明了，对于可分对偶Banach空间$X$上的伴随算子$T$，从任意的可粘递归向量$x_0$，可以构造一个$T$不变的概率测度，它的支持集包含$x_0$.这使我们能够建立一些等价关系，这些运营商，一些强逐点递归概念，在一般情况下是完全不同的。特别是，我们表明，（在我们的框架）迭代复发符合频繁复发;复希尔伯特空间一致复发符合的性质，有一个生成家庭的么模特征向量;和同样的情况下，功率有界算子复自反Banach空间。这些（令人惊讶的）性质很容易推广到乘积和逆动力系统，这意味着与相应的超循环概念的一些关系。最后研究了具有非零粘着回归向量的算子在Baire范畴意义下的典型性。
英文摘要：
We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.
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