Quantitative quasiperiodicity

The Birkhoff ergodic theorem concludes that time averages, i.e. Birkhoff averages, B N ( f ) := Σ n = 0 N − 1 f ( x n ) / N of a function f along a length N ergodic trajectory ( x n ) of a function T converge to the space average ∫ f d μ , where μ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We use a modified average of f ( x n ) by giving very small weights to the ‘end’ terms when n is near 0 or N − 1 . When ( x n ) is a trajectory on a quasiperiodic torus and f and T are C ∞ , our weighted Birkhoff average (denoted W B N ( f ) ) converges ‘super’ fast to ∫ f d μ with respect to the number of iterates N , i.e. with error decaying faster than N − m for every integer m . Our go