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A Quarter Century of Baker-Map Exploration

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Hoover W. G. , Hoover C. G.

2023

tom Vol. 29, No. 1-4

5--16

25 years ago the June 1998 Focus Issue of “Chaos” described the proceedings of a workshop meeting held in Budapest and called “Chaos and Irreversibility”, by the organizers, T. Tél, P. Gaspard, and G. Nicolis. These editors organized the meeting and the proceedings’ issue. They emphasized the importance of fractal structures and Lyapunov instability to modelling nonequilibrium steady states. Several papers concerning maps were presented. Ronald Fox considered the entropy of the incompressible Baker Map B(x, y), shown here in Fig. 1. He found that the limiting probability density after many applications of the map is ambiguous, depending upon the way the limit is approached. Harald Posch and Bill Hoover considered a time-reversible version of a compressible Baker Map, with the compressibility modelling thermostatting. Now, 25 years later, we have uncovered a similar ambiguity, with the information dimension of the probability density giving one value from pointwise averaging and a different one with areawise averaging. Goldstein, Lebowitz, and Sinai appear to consider similar ambiguities. Tasaki, Gilbert, and Dorfman note that the Baker Map probability density is singular everywhere, though integrable over the fractal y coordinate. Breymann, Tél, and Vollmer considered the concatenation of Baker Maps into MultiBaker Maps, as a step toward measuring spatial transport with dynamical systems. The present authors have worked on Baker Maps ever since the 1997 Budapest meeting described in “Chaos”. This paper provides a number of computational benchmark simulations of “Generalized Baker Maps” (where the compressibility of the Map is varied or “generalized”) as described by Kumicák in 2005.