Local stability of ergodic averages
Jeremy D. Avigad, Philipp Gerhardy, Henry Towsner · Transactions of the American Mathematical Society · 2009
We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [ 0 , 1 ] [0,1] and a characteristic function f = χ A f = \chi _A such that the ergodic averages A n f A_n f do not converge to a computable element of L 2 ( [ 0 , 1 ] ) L^2([0,1]) . In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T T on a separable Hilbert space and any element f f , it is possible to compute a bound on the rate of convergence of ⟨ A n f ⟩ \langle A_n f \rangle from T T , f f , and the norm ‖ f ∗ ‖ \| f^* \| of the limit. In particular, if T T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X \mathcal {X} and f