Generic Birkhoff spectra (Q2211118)

Let \(\sigma\) be the one-sided shift on the space \(\Omega=\{0,1\}^{\mathbf N}\). The Birkhoff spectrum of a function \(f\in C(\Omega)\) is the function \[ S_f(\alpha)=\dim_H\left\{\omega\in\Omega:\; \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N} f(\sigma^{n}\omega)=\alpha \right\}, \] where \(\dim_H\) is the Hausdorff dimension. The authors show that for a dense subset of \(C(\Omega)\), the function \(S_f\) is not continuous, while it is continuous for generic \(f\in C(\Omega)\). They also study the one-sided derivatives of \(S_f\) at endpoints of its support.