Restricted Orbit Equivalence for Actions of Discrete Amenable Groups (Cambridge Tracts in Mathematics, Series Number 146)

This 2002 Monograph Offers A Broad Investigative Tool In Ergodic Theory And Measurable Dynamics. The Motivation For This Work Is That One May Measure How Similar Two Dynamical Systems Are By Asking How Much The Time Structure Of Orbits Of One System Must Be Distorted For It To Become The Other. Different Restrictions On The Allowed Distortion Will Lead To Different Restricted Orbit Equivalence Theories. These Include Ornstein's Isomorphism Theory, Kakutani Equivalence Theory And A List Of Others. By Putting Such Restrictions In An Axiomatic Framework, A General Approach Is Developed That Encompasses All These Examples Simultaneously And Gives Insight Into How To Seek Further Applications. The Work Is Placed In The Context Of Discrete Amenable Group Actions Where Time Is Not Required To Be One-dimensional, Making The Results Applicable To A Much Wider Range Of Problems And Examples. Janet Whalen Kammeyer, Daniel J. Rudolph. Includes Bibliographical References And Index. Cover......Page 1 Contents......Page 6 1.1 Overview......Page 8 1.2 A roadmap to the text......Page 10 1.3 History and references......Page 12 1.4 Directions for further study......Page 18 2.1 Orbits, arrangements and rearrangements......Page 20 2.2 Definition of a size and m-equivalence......Page 27 2.3 Seven examples......Page 39 3 The Ornstein-Weiss Machinery......Page 56 4 Copying Lemmas......Page 72 5 m-entropy......Page 98 6.1 Polish topologies......Page 107 6.1.1 Overview of the topology on m-joinings......Page 114 6.2 Modeling pairs of arrangements......Page 115 6.3 Modeling rearrangements......Page 120 6.4 Adding sizes to the picture......Page 136 6.5 More orbit joinings and m-joinings......Page 139 7.1 Perturbing an m-equivalence......Page 146 7.2 The m-distance and m-finitely determined processes......Page 157 7.3 The equivalence theorem......Page 168 Appendix......Page 174 A.1 1-sizes......Page 175 A.2 p-sizes......Page 196 Bibliography......Page 203 Index......Page 206 "This monograph offers a broad investigative tool in ergodic theory and measurable dynamics. The motivation for this work is that one may measure how similar two dynamical systems are by asking how much the time structure of orbits of one system must be distorted for it to become the other.". "Different restrictions on the allowed distortion will lead to different restricted orbit equivalence theories. These include Ornstein's isomorphism theory, the Kakutani equivalence theory and a host of others. By putting such restrictions in an axiomatic framework, a general approach is developed that encompasses all these examples simultaneously and gives insight into how to seek further applications. The work is placed in the context of discrete amenable group actions where time is not required to be one-dimensional, making the results applicable to a much wider range of problems and examples."--BOOK JACKET. This monograph offers a broad investigative tool in ergodic theory and measurable dynamics. The motivation for this work is that one may measure how similar two dynamical systems are by asking how much the time structure of orbits of one system must be distorted for it to become the other. Different restrictions on the allowed distortion will lead to different restricted orbit equivalence theories. These include Ornstein's Isomorphism theory, Kakutani Equivalence theory and a list of others. By putting such restrictions in an axiomatic framework, a general approach is developed that encompasses all of these examples simultaneously and gives insight into how to seek further applications. This 2002 monograph offers a broad investigative tool in ergodic theory and measurable dynamics. The authors develop a general approach and give insights into how to seek further applications. The work is placed in a context that makes the results applicable to a wide range of problems and examples. The purpose of this work is to lift the notion of restricted orbit equivalence to the category of free and ergodic actions of discrete amenable groups.

this Title Offers A Broad Investigative Tool In Ergodic Theory And Measurable Dynamics.