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Preface Introduction A simple system generating a density of states The evolution of densities: an intuitive point of view Trajectories versus densities The toolbox Measures and measure spaces Lebesgue integration Convergence of sequences of functions Markov and Frobenius-Perron operators Markov operators The Frobenius-Perron operator The Koopman operator Studying chaos with densities Invariant measures and measure-preserving transformations Ergodic transformations Mixing and exactness Using the Frobenius-Perron and Koopman operators for cassifying transformations Kolmogorov automorphisms The asymptotic properties of densities Weak and strong precompactness Properties of the averages A_nf Asymptotic periodicity of {P^nf} The existence of stationary densities Ergodicity, mixing, and exactness Asymptotic stability of {P^n} Markov operators defined by a stochastic kernel Conditions for the existence of lower-bound functions The behavior of transformations on intervals and manifolds Functions of bounded variation Piecewise monotonic mappings Piecewise convex transformations with a strong repellor Asymptotically periodic transformations Change of variables Transformations on the real line Manifolds Expanding mappings on manifolds Continuous time systems: an introduction Two examples of continuous time systems Dynamical and semidynamical systems Invariance, ergodicity, mixing, and exactness in semidynamical systems Semigroups of the Frobenius-Perron and Koopman operators Infinitesimal operators Infinitesimal operators for semigroups generated by systems of ordinary differential equations Applications of the semigroups of the Frobenius-Perron and Koopman operators The Hille-Yosida theorem and its consequences Further applications of the Hille-Yosida theorem The relation between the Frobenius-Perron and Koopman operators Discrete time processes embedded in continuous time systems The relation between discrete and continuous time processes Probability theory and Poisson processes Discrete time systems governed by Poisson processes The linear Boltzmann equation: an intuitive point of view Elementary properties of the solutions of the linear Boltzmann equation Further properties of the linear Boltzmann equation Effect of properties of the Markov operator on solutions of the linear Boltzmann equation Linear Boltzmann equation with a stochastic kernel The linear Tjon-Wu equation Entropy Basic definitions Entropy of P^nf when P is a Markov operator Entropy H(P^nf) when P is a Frobenius-Perron operator Behavior of P^nf from H(P^nf) Stochastic perturbation of discrete time systems Independent random variables Mathematical expectation and variance Stochastic convergence Discrete time systems with randomly applied stochastic perturbations Discrete time systems with constantly applied stochastic perturbations Small continuous stochastic perturbations of discrete time systems Stochastic perturbation of continuous time systems One-dimensional Wiener processes (Brownian motion) d-Dimensional Wiener processes (Brownian motion) The stochastic Itô integral: development The stochastic Itô integral: special cases Stochastic differential equations The Fokker-Planck (Kolmogorov forward) equation Properties of the solutions of the Fokker-Planck equation Semigroups of Markov operators generated by parabolic equations Asymptotic stability of solutions of the Fokker-Planck equation An extension of the Liapunov function method References Notation and symbols Index