Spectrum of weighted Birkhoff average

TL;DR

This paper studies the entropy and packing spectra of weighted Birkhoff averages on subshifts of finite type, revealing that the entropy spectrum remains unchanged and characterizing the packing spectrum with two possible cases.

Contribution

It provides a detailed analysis of the entropy and packing spectra of weighted Birkhoff averages, extending understanding of their geometric properties.

Findings

Entropy spectrum of weighted Birkhoff averages equals that of unweighted averages.

Calculated the packing spectrum of weighted Birkhoff averages.

Identified two possible scenarios for the packing dimension of level sets.

Abstract

Let $\{s_n\}_{n\in\N}$ be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average $\frac{1}{S_n}\sum_{k=0}^{n-1}s_k\phi(T^kx)$ on aperiodic irreducible subshift of finite type $\Sigma_{\bf A}$ where $\phi: \Sigma_{\bf A}\mapsto \R$ is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we calculate the packing spectrum of the weighed Birkhoff averages. It turns out that we can have two cases, either the packing dimension of every level set equals to its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.