论文标题
线性动力学系统的复发属性：通过不变测量的方法
Recurrence properties for linear dynamical systems: An approach via invariant measures
论文作者
论文摘要
我们从沿着沿着沿着崇高理论的角度研究线性动力学系统的不同点复发概念。我们表明，从任何重复的经常性向量$ x_0 $中，对于可分开的双Banach Space $ x $的伴随运算符$ t $，一个人可以构造$ t $ invariant概率度量，其中包含$ x_0 $的支持。这使我们能够在某些强烈的重复复发概念之间建立一些等效性，而这些概念通常是完全区分的。特别是，我们表明（在我们的框架中）重复的复发与频繁的复发相吻合。对于复杂的希尔伯特空间，均匀的复发与拥有一个单型特征向量家族的属性相吻合。在复杂的反射巴拉克空间上，由电力型操作员也发生了同样的情况。这些（令人惊讶的）属性很容易被推广到产品和反向动力学系统，这意味着与各自的超循环概念的某些关系。最后，我们研究了在Baire类别意义上，具有非零重复反复的向量的典型运算符。
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.
