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 $\varphi (x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(\varphi )={\sum }_{k=0}^{N-1}(\varphi \circ R_{\alpha }^k)(x)$ with the successive entry $(\varphi \circ R_{\alpha }^N)(x)$. In particular, we are interested in the almost sure limsup behavior of $\frac{(\varphi \circ R_{\alpha }^N)(x)}{S_N(\varphi )(x)}$. 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 $\phi$, where $\phi$ is a smooth non-negative function that has exactly two (non-degenerate) zeros at $\mathbf{p}$ and $\mathbf{q}$. We prove that for a full measure set $(\alpha ,\mathbf{p},\mathbf{q})\in \mathbb{T}\times {\mathbb{T}}^2\times {\mathbb{T}}^2$ the special flow $(T_t)$ exhibits extreme historic behavior proving a conjecture given by Andersson and Guihéneuf.

23 pages, comments welcome