Generic Birkhoff Spectra

Abstract

Suppose that $Ω= \{0, 1\}^ {\mathbb {N}}$ and $ σ$ is the one-sided shift. The Birkhoff spectrum $ \displaystyle S_{f}( α)=\dim_{H}\Big \{ ω\in Ω:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(σ^n ω) = α\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 $L_f = [α_{f, \min}^*, α_{f, \max}^*]$ 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 $f\in C( Ω)$ in the sense of Baire category. For a dense set in $C( Ω)$ the spectrum is not continuous on $ {\mathbb {R}}$, though for the generic $f\in C( Ω)$ 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 $C( Ω)$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.