# American Institute of Mathematical Sciences

December  2020, 40(12): 6649-6679. doi: 10.3934/dcds.2020131

## Generic Birkhoff spectra

 1 Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117, Budapest, Hungary 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Santiago, Chile

* Corresponding author

Received  May 2019 Published  February 2020

Fund Project: 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.

Citation: Zoltán Buczolich, Balázs Maga, Ryo Moore. Generic Birkhoff spectra. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6649-6679. doi: 10.3934/dcds.2020131
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##### References:
An illustration of Remark 5.2.3
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