Observing Ergodic Translations on Compact Abelian Groups with Discontinuous Functions

Abstract

Several authors have considered the problem of global observability for minimal dynamical systems, and particularly the example of ergodic (equivalently, minimal) translations on compact abelian groups. The global observability problem for ergodic translations is considered here in the case where the output function may be discontinuous. The result for continuous output functions, essentially due to McMahon, is described. This result shows that a continuous output function observes all ergodic translations if and only if it has no nontrivial symmetries. This result is extended here in two directions. First, functions continuous except on a meagre set and symmetries modulo meagre sets are considered. A result analogous to the continuous case is obtained that includes and generalizes the continuous case and results of Balogh, Bennett, and Martin on observing ergodic translations with the characteristic functions of certain subsets of the group. Functions continuous except on a set of measure zero are also considered. Secondly, ergodic theory and harmonic analysis techniques are employed to consider the case of intergrable output functions and symmetries modulo sets of measure zero. It is shown that an integrable function observes ergodic translations modulo sets of measure zero if and only if it has no nontrivial symmetries modulo sets of measure zero. The group of symmetries of the output function modulo sets of measure zero can be determined from its Fourier series. The results here show that classical observability can be determined from the Fourier series when the function is continuous almost everywhere, but only observability modulo sets of measure zero can be determined when the output function is merely integrable.