Generic Birkhoff spectra

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.