Speaker
Description
This is a joint work with Zolt\'an Buczolich and Bal\'azs Maga. Let $(\Omega, \sigma)$ be the full-shift of two alphabets, and $f$ be a continuous, real-valued function on it. Let $L_f$ be the set of all of the possible limiting values of the Birkhoff averages of $f$, i.e.
$$L_f := \left\{\alpha \in \mathbb{R} : \exists \, \omega \in \Omega \text{ such that } \lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} f(\sigma^n \omega) = \alpha \right\}.$$
For each $\alpha \in L_f$, we define the level set $$E_f(\alpha) := \left\{\omega \in \Omega: \lim_{N \to \infty} \frac{1}{N}\sum_{n=0}^{N-1} f(\sigma^n \omega) = \alpha\right\},$$ and we define a function $S_f: \mathbb{R} \to \mathbb{R}$, which we refer to as the Birkhoff spectra, as follows:
$$ S_f(\alpha) := \left\{\begin{array}{ll}
\dim_H(E_f(\alpha)) & \alpha \in L_f, \\
0 & \alpha \notin L_f,
\end{array}\right.$$
where $\dim_H$ is the Hausdorff dimension.
In this talk, we will discuss shapes and properties of the Birkhoff spectrum $S_f$ for generic/typical continuous functions $f$ in the sense of Baire category. In particular, we will be interested in the behavior of the spectrum near the boundary of $L_f$, such as the continuity and the values of one-sided derivatives.
For more information, please refer to: \href{https://arxiv.org/abs/1905.06001}{arXiv:1905.06001}.
