Suppose that $ \Omega = \{0, 1\}^\mathbb{N} $ and $ \sigma $ is the one-sided shift. The Birkhoff spectrum $ S_{f}(α) = \dim_{H}\Big \{ \omega\in\Omega:\lim\limits_{N \to \infty} \frac{1}{N} \sum\limits_{n = 1}^N f(\sigma^n \omega) = \alpha \Big \}, $ where $ \dim_{H} $ is the Hausdorff dimension. It is well-known that the support of $ S_{f}(α) $ is a bounded and closed interval $ L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*] $ and $ S_{f}(α) $ on $ L_{f} $ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $ f\in C(\Omega) $ in the sense of Baire category. For a dense set in $ C(\Omega) $ the spectrum is not continuous on $ \mathbb{R} $, though for the generic $ f\in C(\Omega) $ the spectrum is continuous on $ \mathbb{R} $, but has infinite one-sided derivatives at the endpoints of $ L_{f} $. We give an example of a function which has continuous $ S_{f} $ on $ \mathbb{R} $, but with finite one-sided derivatives at the endpoints of $ L_{f} $. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $ f $ and $ g $ are close in $ C(\Omega) $ then $ S_{f} $ and $ S_{g} $ are close on $ L_{f} $ apart from neighborhoods of the endpoints.

中文翻译：

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通用伯克霍夫谱

假设$ \ Omega = \ {0，1 \} ^ \ mathbb {N} $和$ \ sigma $是单向移位。Birkhoff谱$ S_ {f}（α）= \ dim_ {H} \ Big \ {\ omega \ in \ Omega：\ lim \ limits_ {N \ to \ infty} \ frac {1} {N} \ sum \ limits_ {n = 1} ^ N f（\ sigma ^ n \ omega）= \ alpha \ Big \}，其中$ \ dim_ {H} $是Hausdorff维数。众所周知，$ S_ {f}（α）$的支持是一个有界的封闭区间$ L_f = [\ alpha_ {f，\ min} ^ *，\ alpha_ {f，\ max} ^ *] $和L_ {f} $上的$和$ S_ {f}（α）$是凹且上半连续的。我们对频谱的可能形状/特性感兴趣，尤其是对于Baire类别中C（\ Omega）$中的通用/典型$ f \。对于C（\ Omega）$中的密集集合，频谱在$ \ mathbb {R} $上不是连续的，尽管对于C（\ Omega）$中的泛型$ f \，频谱在$ \ mathbb {R } $，但是在$ L_ {f} $的端点处具有无限的单边导数。我们给出一个函数示例，该函数在$ \ mathbb {R} $上具有连续的$ S_ {f} $，但是在$ L_ {f} $的端点处具有有限的单侧导数。该功能的频谱可以尽可能接近“最小频谱”。我们使用以下假设：如果两个函数$ f $和$ g $在$ C（\ Omega）$中接近，则$ S_ {f} $和$ S_ {g} $在$ L_ {f} $上接近，而附近端点。