Generic Birkhoff Spectra

Suppose that $\mathrm{\Omega }=\{0,1{\}}^{\mathbb{N}}$ and $\sigma$ is the one-sided shift. The Birkhoff spectrum $ \displaystyle S_{f}( {\alpha})=\dim_{H}\Big { {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big },$where$\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_f(\alpha )$ is a bounded and closed interval $L_f=[{\alpha }_{f,\min }^{\ast },{\alpha }_{f,\max }^{\ast }]$ and $S_f(\alpha )$ on $L_f$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C(\mathrm{\Omega })$ in the sense of Baire category. For a dense set in $C(\mathrm{\Omega })$ the spectrum is not continuous on $\mathbb{R}$, though for the generic $f\in C(\mathrm{\Omega })$ 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$S_{g}$arecloseon$L_{f}$ apart from neighborhoods of the endpoints.