Happy Birthday, Ed Lorenz
In honor of Ed Lorenz’s birthday, I want to share a little adventure I took in the nonlinear dynamics literature after finding an error. The error resulted from someone using Wolfram’s MathWorld as a reference for the Kolmogorov-Sinai entropy. I managed to track down the origin of the error (I think) and sent the correction along to MathWorld.
My message to MathWorld is below.
Error in Kolmogorov Entropy Entry on MathWorld:
The definition you give is not for the Kolmogorov-Sinai / metric entropy. You cite Schuster/Wolfram and Ott. Ott doesn’t contain this definition, so you must be referencing Schuster/Wolfram. Schuster/Wolfram gives this definition on page 96 of their text, citing Farmer 1982a/b. But neither Farmer paper gives this definition for the Kolmogorov-Sinai entropy. Both Farmer 1982a and 1982b give the standard definition, in terms of the supremum of the entropy rate of the symbolized dynamics over all finite partitions of the state space. The best guess I could find for the origin of the Schuster/Wolfram error was in Grassberger 1983, which cites Farmer 1982a, and gives the incorrect definition for the Kolmogorov-Sinai entropy that you provide. Interestingly, the Grassberger paper also gets the page number wrong for Farmer 1982a: citing page 566 rather than 366.
Thanks for the chance to go on this mathematical excavation!
Farmer 1982a: J. Doyne Farmer. "Chaotic attractors of an infinite-dimensional dynamical system." Physica D: Nonlinear Phenomena 4.3 (1982): 366-393.
Farmer 1982b: J. Doyne Farmer. "Information dimension and the probabilistic structure of chaos." Zeitschrift für Naturforschung A 37.11 (1982): 1304-1326.
Grassberger 1983: Peter Grassberger, and Itamar Procaccia. "Estimation of the Kolmogorov entropy from a chaotic signal." Physical Review A 28.4 (1983): 2591.