Article Contents
Article Contents

# Generic Birkhoff spectra

• * Corresponding author

ZB was supported by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124003.
BM was supported by the ÚNKP-18-2 New National Excellence of the Hungarian Ministry of Human Capacities, and by the Hungarian National Research, Development and Innovation Office–NKFIH, Grant 124749.
RM was partially supported by CONICYT-FONDECYT Postdoctorado 3170279

• 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.

Mathematics Subject Classification: Primary: 37A30; Secondary: 28A80, 37B10, 37C45.

 Citation:

• Figure 1.  An illustration of Remark 5.2.3

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