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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 362, Number 1, January 2010, Pages 261288
S 00029947(09)048144
Article electronically published on July 31, 2009
LOCAL STABILITY OF ERGODIC AVERAGES
JEREMY AVIGAD, PHILIPP GERHARDY, AND HENRY TOWSNER
Abstract. 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 transfor
mation of [0, 1] and a characteristic function f = A such that the ergodic
averages Anf do not converge to a computable element of L2([0, 1]). In par
ticular, there is no computable bound on the rate of convergence for that
sequence. On the other hand, we show that, for any nonexpansive linear op
erator T on a separable Hilbert space and any element f, it is possible to
compute a bound on the rate of convergence of Anf from T, f, and the
norm f of the limit. In particular, if T is the Koopman operator arising
from a computable ergodic measure preserving transformation of a probability
space X and f is any computable element of L2(X), then there is a computable
