Capturing deviation from ergodicity at different scales

We address here the issue of quantifying the extent to which a given dynamical system falls short of being ergodic and introduce a new multiscale technique which we call the “ergodicity defect”. Our approach is aimed at capturing both deviation from ergodicity and its dependence on scale. The method uses ergodic theory of dynamical systems and applies harmonic analysis, in particular the scaling analysis is motivated by wavelet theory.We base the definition of the ergodicity defect on the Birkhoff characterization. We systematically exploit the role of the observation function by using characteristic functions arising from a dyadic equipartition of the phase space. This allows us to view the dependence of the defect on scale. In order to build intuition, we consider the defect for specific examples with known dynamic properties and we are able to explicitly compute the defect for some of these simple examples. We focus on three distinctive cases of the dependence of the defect on scale: (1) a defect value that increases as the scale becomes finer, (2) a defect value decreasing with scale and (3) a defect value independent of scale, which occurs for instance when a map is ergodic. We explain the information contained in these three scenarios. We see more complicated behavior with an example which has invariant subsets at various scales.