Probabilistic Properties of Deterministic SystemsThis book shows how densities arise in simple deterministic systems. There has been explosive growth in interest in physical, biological and economic systems that can be profitably studied using densities. Due to the inaccessibility of the mathematical literature there has been little diffusion of the applicable mathematics into the study of these 'chaotic' systems. This book will help to bridge that gap. The authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial differential equations. They have drawn examples from many scientific fields to illustrate the utility of the techniques presented. The book assumes a knowledge of advanced calculus and differential equations, but basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed. |
Contents
The toolbox | 13 |
Studying chaos with densities | 45 |
classifying transformations | 63 |
The asymptotic properties of densities | 77 |
The behavior of transformations | 114 |
an introduction | 163 |
Discrete time processes embedded | 219 |
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Probabilistic Properties of Deterministic Systems Andrzej Lasota,Michael C. Mackey Limited preview - 2008 |
Common terms and phrases
applied arbitrary assume behavior Borel bounded calculate called chapter completes concept condition consequence consider constant contains continuous convergence Corollary corresponding defined definition denotes derivative differential equations dynamical system easy entropy equal ergodic exact examine example exists fact Figure Finally finite fixed Frobenius-Perron operator function Further given gives Hence holds implies important independent inequality initial integral interval introduced invariant limit linear mapping Markov operator means measure space mixing Observe obtain Pf(x positive possible probability problem proof properties Proposition prove random variables Remark respect satisfies semigroup sequence shown simple solution stationary density stochastic stochastic differential equations stochastic kernel strong strongly subset sufficient term Theorem trajectory transformation unique weakly write zero