Lasota A., Mackey M. Probabilistic properties of deterministic systems

Cambridge: Cambridge University Press, 1985. — 366 p.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 methodNotation and symbols