Generic Birkhoff Spectra

Abstract

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ 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 $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$ 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 $f\in C( {\Omega})$ 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.