标题： 矩阵的伪谱导致的繁荣与萧条周期 显示英文标题

标题： Boom and bust cycles due to pseudospectra of matrices with unimodular spectra

摘要： 我们讨论通过增加非正规矩阵的幂次得到的动力学，这些矩阵是单位矩阵的根，因此所有特征值都在单位圆上。 直觉上，人们会认为随着幂次的增加，这些幂的期望值不会增长。 我们证明，相反，这种出乎意料的行为是可能的。 在矩阵无限大的极限情况下，可以出现指数增长。 对于有限矩阵，这种指数增长是指数增长和指数衰减重复循环的一部分。 当谱与伪谱不同时，这种效应可能发生，指数增长速率由伪谱给出。 我们表明，这种效应出现在二维非相互作用系统研究中出现的一类转移矩阵中，出现在描述埃伦费斯特瓮的矩阵中，以及在阶梯随机电路中之前观察到的纯度动力学中。 显示英文摘要

摘要： We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.