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Ergodic Theory, Randomness, and "Chaos"

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Ergodic Theory, Randomness, and 8. S. P. Havera and F. C. Bellrose, Wetlands 4, 19 (1985). 19. W. C. Johnson and D. M. Sharpe, For. Sci. 22, 307 (1976). 9. F. H. Nichols, J. E. Cloern, S. N. Luoma, D. H. Peterson, Science 231, 567 20. M. G. Turner and C. L. Ruscher, Landscape Ecol. 1, 241 (1988). (1986). 21. M. G. Turner, Environ. Manage. 11, 237 (1987). 10. F. C. White, J. R. Hairston, W. N. Musser, H. F. Perkins, J. F. Reed, J. Soil Water 22. G. W. Johnson, The Wasted Land (Univ. of North Carolina Press, Chapel Hill, Conserv. 36, 172 (1981). 1938). 11. P. Crosson, Conservation Tillage and Conventional Tillage: A Comparative Assessment 23. J. S. Olson, in Productivity of World Ecosystems (National Academy of Sciences, (Soil Conscrvation Society of America, Ankeny, IA 1981); R. E. Sijka, G. W. Washington, DC, 1975); R. H. Waring and W. H. Schlesinger, Forest Ecosystems, Langdale, D. L. Karlen, Adv. Agron. 37, 155 (1984). Conservation tillage reduces Concepts and Management (Academic Press, New York, 1985). soil loss and surface runoff of nitrogen, but it is not certain if leaching and 24. See reviews by H. T. Odum, H. E. Daly, R. Costanza, and others [in Integration of contamination of ground water by nitrogen is also reduced [T. J. Logan, J. W. Economy and Ecology, A. M. Jansson, Ed. (Univ. of Stockholm Press, Stockholm, Davidson, J. L. Baker, M. R. Overcash, Eds., Effects of Conservation Tillage on Sweden, 1984)]. Groundwater Quality: Nitrates and Pesticides (Lewis, Chelsea, MI, 1987)]. 25. M. G. Turner, E. P. Odum, R. Costanza, T. P. Springer, Environ. Manage. 12, 237 12. S. Postel, in The State of the World, L. Brown, Ed. (World Watch Institute, (1988). Washington, DC, 1986), chap. 3. 26. K. Parker, E. P. Odum, J. L. Cooley, Natural Ecosystems ofthe Sunbelt (Institute of 13. M. R. Gebhardt, T. C. Daniel, E. E. Schweizer, R. R. Allmarus, Science 230, 625 Ecology, University of Georgia, Athens, 1981). (1985). 27. D. F. Spencer, S. B. Alpert, H. H. Gilman, Science 232, 609 (1986). 14. P. F. Hendrix et al., BioScience 36, 374 (1987). 28. E. P. Odum, BioScience 34, 558 (1984). 15. R. L. Blevins, M. S. Smith, G. W. Thomas, in No-Tillage Agriculture: Principles and 29. H. W. Odum, Southem Regions ofthe United States (Univ. of North Carolina Press, Practices, R. E. Phillips and S. H. Phillips, Eds. (Van Nostrand Reinhold, New Chapel Hill, 1936). York, 1984); J. W. Doran, Soil Sci. Soc. Am. J. 44, 765 (1980). 30. E. P. Odum,J. Soil Water Conserv. 42, 412 (1987). 16. G. J. House, B. R. Stinner, D. A. Crossley, E. P. Odum,J. Appl. Ecol. 21, 991 31. I am indebted to M. G. Turner, a research associate on the Georgia Project, which (1984); B. R. Stinner, D. A. Crossley, E. P. Odum, R. L. Todd, Ecology 65, 354 was supported by the W. K. Kellogg Foundation. Other members ofthe task force (1984). were E. 0. Box, J. L. Cooley, F. B. Golley, J. E. Kundell, J. A. Little, V. 17. E. P. Odum and M. G. Turner, The Georgia Landscape: A Changing Resource Mcentemeyer, D. Morrison, H. R. Pulliam, and R. Westmacot. I also acknowlcdge (Institute of Ecology, University of Georgia, Athens, 1988). work ofD. A. Crossley, D. C. Colman, P. F. Hendrix, B. R. Stinner, G. J. Housc, 18. , in Trends in Landscape Ecology, I. S. Zonneveld and R. T. T. Forman, Eds. and P. M. Groffman of the agroecosystem team, supported by grants from the (Springer-Verlag, New York, in press). National Science Foundation. .l Ergodic Theory, Randomness, and "Chaos" D. S. ORNSTEIN on August 15, 2011 way (in other words, maintain the correspondences for all time) (1). Ergodic theory is the theory of the long-term statistical If we preserved the topology instead of probabilities, we would be behavior of dynamical systems. The baker's transforma- studying the qualitative theory of-ordinary differential equations tdon is an object of ergodic theory that provides a para- initiated by Poincare. www.sciencemag.org digm for the possibility ofdeterministic chaos. It can now Abstract dynamical systems are natural objects from the mathe- be shown that this connection is more than an analogy matical point ofview, and they arise in many different contexts (even and that at some level of abstraction a large number of in areas as far afield as number theory); elucidating their structure is systems governed by Newton's laws are the same as the considered an important mathematical problem. Much ofthe work baker's transformation. Going to this level of abstraction in ergodic theory has had little to do with its initial motivation, but helps to organize the possible kinds ofrandom behavior. recently certain problems, some of which had been unsolved for The theory also new concrete results. For example, more than a decade, have been solved, and a group of results has gives Downloaded from one can show that the same process could be produced by been obtained that does relate to concrete systems such as the a mechanism governed by Newton's laws or by a mecha- billiard system. These results, which I will refer to as isomorphism nism governed by coin tossing. It also gives a statistical theory, center around a better understanding of a certain abstract analog of structural stability. dynamical system called the baker's transformation. This is a map of the unit square onto itself (Fig. 1). We first stretch the square, doubling its width and halving its height. We next stack the right halfofthe elongated rectangle above the left half (the shape is again E 1 RGODIC THEORY AROSE OUT OF AN ATrEMPT TO UNDER- a square). These two steps give the baker's transformation. Iflooked stand the long-term statistical or probabilistic behavior of at properly the baker's transformation can serve as the mathematical dynamical systems such as the motions ofa billiard ball or the model for coin tossing (this is easy to see), and in this sense is motions of the earth's atmosphere. The theory focused on certain completely random while it is deterministic in the sense that every mathematical objects called abstract dynamical systems or measure- point moves in a definite way. preserving flows. The idea here is to abstract out the statistical The baker's transformation is often used as a paradigm for properties and ignore other properties of the dynamical system. explaining the possibility of deterministic chaos (2, 3), that is, Thus two systems are considered the same when viewed as an systems that evolve according to Newton's laws but nevertheless abstract system (we call these isomorphic) if, after we ignore sets (or appear to be random. In recent years people have become increas- events) of probability zero, there is a one-to-one correspondence ingly aware of the ubiquity of this phenomenon. between the points in their phase spaces (see below) so that corresponding sets have the same probability and evolve in the same The author is professor of mathematics at Stanford University, Stanford, CA 94305. I82 SCIENCE, VOL. 243 our random system could be produced by Newton's laws or by coin flipping. This article has a dual purpose. I would like to explain some results in ergodic theory by describing what they say in a simple Fig. 1. The bakers transformation. concrete context. I also hope that these results will shed some light on the nature ofchaos. For a more complete technical version ofthis New results follow from isomorphism theory: article, see (7), which is based largely on (8). 1) At the level ofabstraction ofisomorphism there is a unique (4, 5) system that is the most random possible. I will refer to it as the Bernoulli flow or B, The Phase of a System Even though "most random" is not well defined, I will explain Space Dynamical some precise results that will make the above statement clear. It is Consider the example of a billiard ball moving on a rectangular important, however, to identify isomorphic systems and, unless we table with a convex obstacle. The ball moves in a straight line at unit do this, "most random" has no meaning. Because there is no agreed speed and bounces off the obstade and the edges with the angle of definition of "chaotic" in the literature on chaos, result 1 gives us reflection equal to the angle of incidence. something definitive to aim at when studying specific systems. The configuration of the system is completely determined by the 2) If we look at B,, in other words, a system isomorphic to B,, at position and velocity of the ball. The phase space is the set of all regular time intervals (for example, every hour on the hour), then configurations of the system. Each point (configuration) in the this new system is isomorphic to the baker's transformation. phase space moves along a well-defined trajectory or orbit in the 3) There are systems that obey Newton's laws that are isomor- phase space. phic to Bt. A billiard system with one or several convex obstacles is Phase space also has a probability structure: every reasonable (in an example (without the obstade it is not chaotic). Other examples other words, measurable) set in phase space is assigned a probabili- ofBt are geodesic flow on a manifold of negative curvature and the ty-the probability offinding the system in that set. The probability Lorenz attractor. Dissipative examples ofBt are provided by Smale's structure has the property that, as a set evolves in time, its axiom A attractors. Isomorphism theory provides the only method probability does not change.
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