We study the properties of $ \Phi $-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $ \Phi $-irregular set in terms of entropy on chain recurrent classes and prove that $ \Phi $-irregular sets of full entropy are typical. We also consider $ \Phi $-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

中文翻译：

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关于熵 \ begin {document} $ \ Phi $ \ end {document}-不规则和 \ begin {document} $ \ Phi $ \ end {document}带有阴影属性的高级别集

我们研究具有阴影特性的动力学系统中$ \ Phi $-不规则集（Birkhoff平均值偏离的点集）的属性。我们根据链递归类上的熵估计$ \ Phi $-不规则集的拓扑熵，并证明$ \ Phi $-不规则集的全熵是典型的。我们还考虑$ \ Phi $级集（Birkhoff平均值在指定间隔内的点集），将它们携带的熵与某些遍历测度的熵相关联。最后，我们考虑参考标准的水平集来研究较大偏差的问题。