Koopman operators and measures of maximal entropy via symbolic dynamics

In our previous work, we introduced symbolic Extended Dynamic Mode Decomposition (EDMD), a method combining EDMD with symbolic dynamics to estimate the Koopman operator using dictionaries constructed from cylinder sets of a generating partition. That work required knowledge of an invariant measure and was validated on systems with known partitions. In this paper, we make two significant advances. First, we prove that our method can be performed with no knowledge of any invariant measures for the system, allowing any sampling method that can see all open sets. Second, we prove that a byproduct of our method is a sequence of measures μm, which approximate the measure of maximal entropy (MME) so long as the dynamical system is equivalent to a Markov or Sofic symbolic shift. We present numerical results on three newly tested dynamical systems: the Liverani-Saussol-Vaienti map, a non-Markov piecewise linear map, and the Arnold Cat Map. We obtain good results for the estimation of spectral data and the MME for each system. In addition, we present some new data-driven methods for estimation of symbolic cylinder sets for the Cat Map.