The influence of the maximal summand on ergodic sums of non-integrable observables over rotations

TL;DR

This paper investigates the asymptotic behavior of ergodic sums of a specific non-integrable observable over irrational rotations, revealing how Diophantine properties influence the limsup behavior and its implications for special flows with extreme historic behavior.

Contribution

It establishes a link between Diophantine properties of rotation angles and the limsup behavior of ergodic sums, and demonstrates the occurrence of extreme historic behavior in certain special flows.

Findings

Limsup of normalized observable equals 0 or infinity depending on Diophantine properties.

Set of angles with limsup zero has Hausdorff dimension 1/2.

Full measure of parameters leads to extreme historic behavior in the flow.

Abstract

For $R_{\alpha}$ being an irrational rotation of angle $\alpha$ on the one torus $\mathbb{T}$ and $\phi(x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(\phi)=\sum_{k=0}^{N-1}(\phi\circ R_{\alpha}^k)(x)$ with the successive entry $(\phi\circ R_{\alpha}^N)(x)$. In particular, we are interested in the almost sure limsup behavior of $\frac{(\phi\circ R_{\alpha}^N)(x)}{S_N(\phi)(x)}$. We show that depending on the Diophantine properties of $\alpha$ we have that the limsup either equals $0$ or $\infty$. Moreover, we show that those $\alpha$ for which the limsup equals $0$ form an atypical set in the sense that its Hausdorff dimension equals $\frac{1}{2}$. These results have consequences in studying a reparametrization $(T_t)$ of the linear flow $(L_t)$ with direction $(1,\alpha)$ on the two torus $\mathbb{T}^2$ with function $\varphi$, where $\varphi$ is a smooth non-negative function that has exactly two (non-degenerate) zeros at $\bf p$ and $\bf q$. We prove that for a full measure set $(\alpha, {\bf p}, {\bf q})\in \mathbb{T}\times \mathbb{T}^2\times \mathbb{T}^2$ the special flow $(T_t)$ exhibits extreme historic behavior proving a conjecture given by Andersson and Guih\'eneuf.