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lecture-25 - Chapter 25 Ergo dicity This lecture explains...

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Chapter 25 Ergodicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these proper- ties (especially for AMS processes), and deduces some consequences. The most important one is that sample averages have deterministic limits. 25.1 Ergodicity and Metric Transitivity Definition 300 A dynamical system Ξ , X , μ, T is ergodic , or an ergodic system or an ergodic process when μ ( C ) = 0 or μ ( C ) = 1 for every T -invariant set C . μ is called a T -ergodic measure , and T is called a μ -ergodic transformation, or just an ergodic measure and ergodic transformation , respectively. Remark: Most authorities require a μ -ergodic transformation to also be measure-preserving for μ . But (Corollary 54) measure-preserving transforma- tions are necessarily stationary, and we want to minimize our stationarity as- sumptions. So what most books call “ergodic”, we have to qualify as “stationary and ergodic”. (Conversely, when other people talk about processes being “sta- tionary and ergodic”, they mean “stationary with only one ergodic component”; but of that, more later. Definition 301 A dynamical system is metrically transitive , metrically inde- composable , or irreducible when, for any two sets A, B X , if μ ( A ) , μ ( B ) > 0 , there exists an n such that μ ( T - n A B ) > 0 . Remark: In dynamical systems theory, metric transitivity is contrasted with topological transitivity: T is topologically transitive on a domain D if for any two open sets U, V D , the images of U and V remain in D , and there is an n such that T n U V = . (See, e.g., Devaney (1992).) The “metric” in “metric transitivity” refers not to a distance function, but to the fact that a measure is involved. Under certain conditions, metric transitivity in fact 167
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CHAPTER 25. ERGODICITY 168 implies topological transitivity: e.g., if D is a subset of a Euclidean space and μ has a positive density with respect to Lebesgue measure. The converse is not generally true, however: there are systems which are transitive topologically but not metrically. A dynamical system is chaotic if it is topologically transitive, and it contains dense periodic orbits (Banks et al. , 1992). The two facts together imply that a trajectory can start out arbitrarily close to a periodic orbit, and so remain near it for some time, only to eventually find itself arbitrarily close to a di ff erent periodic orbit. This is the source of the fabled “sensitive dependence on ini- tial conditions”, which paradoxically manifests itself in the fact that all typical trajectories look pretty much the same, at least in the long run. Since metric transitivity generally implies topological transitivity, there is a close connection between ergodicity and chaos; in fact, most of the well-studied chaotic systems are also ergodic (Eckmann and Ruelle, 1985), including the logistic map. How- ever, it is possible to be ergodic without being chaotic: the one-dimensional rotations with irrational shifts are, because there periodic orbits do not exist, and a fortiori are not dense.
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