Jane Hawkins - Ergodic Dynamics: From Basic Theory to Applications

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The word ergodic is one of many scientific portmanteau words that were assembled from Greek words, in this case by mathematical physicists. They produced a new label for a type of dynamical behavior exhibiting some uniform randomness. Words for work (ergon) and path (odos) were combined to give ergodic; but, why was that meaningful? To make sense of work-path, we consider a system of many moving particles, such as a fluid, with the property we can understand the entire system reasonably well by measuring and averaging the work done along just one randomly chosen orbit path. Then, there is a certain intrinsic randomness exhibited by the dynamical system, since we do not know in advance which initial point of the more than 1027 possibilities to follow; we call this system ergodic. Another way to think of an ergodic dynamical system is to imagine that any randomly chosen point has an orbit that passes through a neighborhood of every possible state of the system, spending the right proportion of its time there through its recurring visits. Therefore, following the path of one point tells you about the entire system. Unfortunately, not all dynamical systems have this indecomposability to them, and not every point in an ergodic system will unlock the behavior of the whole system. It is in understanding whys or why nots, the basic examples, and stronger related properties that we get into the beautiful mathematics of the subject of ergodic theory.

The term ergodic was coined by Boltzmann in the late 1860s in the context of the statistical mechanics of gas particles; it is relevant that he was wrong, or at least overly hopeful in his original conjecture that every classical system of interest was ergodic. The term was subsequently adopted by both mathematicians and physicists, its meaning bifurcated and mutated over the decades, and it currently means slightly different things to mathematicians, applied mathematicians, and physicists. Vocabulary that comes into existence in this contrived way frequently leaves most readers out in the cold. One goal of this text is to show that the mystery surrounding ergodic theory is unwarranted. The subject could just as easily be called dynamical systems, except that studying the subject using only topology and calculus does not capture the essence of the probabilistic randomness involved in an ergodic system. We study topological dynamical systems in this book too, as the interplay between the topological and statistical properties in many physical and natural examples is what lends so much beauty to the subject. In fact, from the start, we give all of our dynamical systems both topological and measurable structure to avoid deciding which tool kit we can use; we hope to show the reader how to use both interchangeably.