Speaker
Description
We consider an irrational rotation R_α on the torus (R_α(x)= x+α mod 1) and the non-integrable observable f(x)=1/x-1/(1-x) and study the Birkhoff sum S_n=f+...+f(R_α^(n-1)). As by Aaronson's theorem it is not possible to obtain strong convergence for S_n/d_n where d_n is any norming sequence, we look at the trimmed sum where we delete one or more of the maximal terms to obtain strong convergence. Depending on the Diophantine properties of α we may need to delete only one or many of the maximal entires of the Birkhoff sum to obtain strong convergence. Moreover, we look at extravagance, i.e. the limsup behaviour of f(R_α^n) / S_n and show under which conditions it equals ∞ or 0. These results have consequences in studying a reparametrization (T_t) of the linear flow (L_t) with direction (1,α) on the two torus T2 with function φ, where φ is a smooth non-negative function that has exactly two (non-degenerate) zeros at p and q. We prove that for a full measure set (α,p,q)∈ T×T^2×T^2 the special flow (T_t) exhibits extreme historic behavior proving a conjecture given by Andersson and Guihéneuf. The talk is based on joint work with Max Auer and with Adam Kanigowski.