Spectrum of weighted Birkhoff average
Volume 269 / 2023
Studia Mathematica 269 (2023), 65-82
MSC: Primary 28A80; Secondary 28A78, 37D20, 40G05.
DOI: 10.4064/sm210908-19-6
Published online: 3 October 2022
Abstract
Let $\{s_n\}_{n\in \mathbb N}$ be a decreasing nonsummable sequence of positive reals. We investigate the weighted Birkhoff average $\frac {1}{S_n}\sum _{k=0}^{n-1}s_k\phi (T^kx)$ on an aperiodic irreducible subshift $\Sigma _{\bf A}$ of finite type where $\phi : \Sigma _{\bf A}\to \mathbb R$ is a continuous potential. Firstly, we show that the entropy spectrum of the weighted Birkhoff averages remains the same as that of the Birkhoff averages. Then we calculate the packing spectrum of the weighted Birkhoff averages. It turns out that we can have two cases: either the packing dimension of every level set equals its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.
