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  • This dissertation develops a functional-analytic framework for discrete-time dynamical systems by studying Koopman and Perron-Frobenius operators in both topological and measure-preserving settings.It establishes new operator-theoretic characterizations of geometric properties of the underlying dynamics, including injectivity, density of image, surjectivity, and homeomorphism, and connects these to ergodic decompositions and mixing behavior.A central contribution is the analysis of entropy through operator approximation: the work relates measure-theoretic entropy, Voiculescu’s approximation entropy, and newly introduced fractional approximation bounds, showing in particular that positive measure-theoretic entropy enforces unbounded approximation complexity.From a data-driven perspective, the thesis studies extended Dynamic Mode Decomposition and proves quantitative lower bounds on the dimension required for accurate finite-time Koopman prediction, with linear and exponential growth laws governed by approximation and measure entropy.Finally, it introduces Banach lattice entropy as a unifying notion recovering both Kolmogorov-Sinai and topological entropy, and characterizes uniform ergodicity of Koopman operators in terms of eventual periodicity (topological case) and periodicity (measure-preserving case).
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