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The operator theoretic framework for dynamical systems studies the dynamics induced on some functional space like $C^0(M)$ or $L^2(\mu)$, instead of the trajectories on the underlying phase space $M$. It transforms the dynamics under any nonlinear flow $\Phi^t$ into a linear map on some Banach / Hilbert space. The dynamics is induced by the Koopman operator $U^t$, which acts on functions by time shifts, namely, $(U^t f)(x) = f (\Phi^t x)$. The operator theoretic framework offers an alternative way of restating several questions in dynamics, and the spectral properties of $U^t$ have important implications on the actual dynamics. I will discuss the connections of $U^t$ with the statistical / ergodic properties of the underlying dynamics. I will also discuss how these operators can be well approximated by matrices, and discuss various convergence results. |