Overview

Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note [44] suggested to study ergodic theorems for the more general situation when one applies in turn different transforma­ tions chosen at random. Their program was fulfilled by S. Kakutani [23] in 1951. 'Both papers considered the case of transformations with a common invariant measure. Recently Ohno [38] noticed that this condition was excessive. Ergodic theorems are just the beginning of ergodic theory. Among further major developments are the notions of entropy and characteristic exponents. The purpose of this book is the study of the variety of ergodic theoretical properties of evolution processes generated by independent applications of transformations chosen at random from a certain class according to some probability distribution. The book exhibits the first systematic treatment of ergodic theory of random transformations i.e., an analysis of composed actions of independent random maps. This set up allows a unified approach to many problems of dynamical systems, products of random matrices and stochastic flows generated by stochastic differential equations.

Full Product Details

Author:   Yuri Kifer
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   Softcover reprint of the original 1st ed. 1986
Volume:   10
Dimensions:   Width: 15.20cm , Height: 1.20cm , Length: 22.90cm
Weight:   0.334kg
ISBN:  

9781468491777

ISBN 10:   1468491776
Pages:   210
Publication Date:   02 June 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

I. General analysis of random maps.- 1.1. Markov chains as compositions of random maps.- 1.2. Invariant measures and ergodicity.- 1.3. Characteristic exponents in metric spaces.- II. Entropy characteristics of random transformations.- 2.1. Measure theoretic entropies.- 2.2. Topological entropy.- 2.3. Topological pressure.- III. Random bundle maps.- 3.1. Oseledec’s theorem and the “non-random” multiplicative ergodic theorem.- 3.2. Biggest characteristic exponent.- 3.3. Filtration of invariant subbundles.- IV. Further study of invariant subbundles and characteristic exponents.- 4.1. Continuity of invariant subbundles.- 4.2 Stability of the biggest exponent.- 4.3. Exponential growth rates.- V. Smooth random transformations.- 5.1. Random diffeomorphisms.- 5.2. Stochastic flows.- A. 1. Ergodic decompositions.- A.2. Subadditive ergodic theorem.- References.

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