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

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.