论文标题
转移算子中的一般渐近扰动理论
General asymptotic perturbation theory in transfer operators
论文作者
论文摘要
我们研究了在扰动的转移操作员，相应特征功能和双重操作员的相应特征向量的特征值中特征值的高阶渐近膨胀。在我们的主要结果中，我们给出了这些系数的明确表达，并且在线性操作员的轻度条件下（甚至没有规范）线性空间。此外，我们研究了在弱条件下剩余的足够条件，从某种意义上说，扰动操作员的特征值没有均匀的频谱间隙。作为我们的主要应用，我们将结果应用于与所谓图形的Markov Systems的极限集相关的Gibbs度量的渐近行为。此外，我们证明了Ruelle转移算子的渐近行为，没有均匀的光谱间隙。在另一个例子中，我们提到，在Gouëzel和Liverani给出的抽象渐近理论条件下以及在轻度条件下，我们的主要结果的条件得到了满足。
We study higher-order asymptotic expansions of eigenvalues in perturbed transfer operators, of the corresponding eigenfunctions and of the corresponding eigenvectors of the dual operators. In our main result, we give explicit expressions of these coefficients and these remainders under mild conditions of linear operators and of (there is even no norm) linear spaces. Moreover we investigate sufficient conditions for convergence of the remainders under weak conditions in the sense that the eigenvalue of perturbed operators does not have a uniform spectrum gap. As our main application, we apply our result to the asymptotic behaviour of Gibbs measures associated with the Hausdorff dimension of the limit set of the so-called graph directed Markov systems. Moreover, we demonstrate the asymptotic behaviours of Ruelle transfer operators without a uniform spectral gap. In another example, we mention that the conditions of our main results are satisfied under the conditions of abstract asymptotic theory given by Gouëzel and Liverani and under mild conditions.
