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Thursday, November 05, 2009

Properties of the WTS II - Addendum

Properties of the Wine Tasting Sequence II - Addendum wb 091105

Properties of the Wine Tasting Sequence II Sctn. 7 described the first of an infinite class of homomorphic unfolding sequences, all having fascinating properties, generated by functions f satisfying:

7.2.4. f((&f^n)(s)) = s, n = 0,1,2,...

Here is the second, again using dots for cosmetic punctuation:

7.2.5.1. S2 = 0.1.2.0.01.012.0120.012001.012001012.0120010120120.0120010120120012001...

So that you can grasp its gestalt visually, here it is without punctuation:

7.2.5.2. S2 = 012001012012001200101200101201200101201200120010120120012001...

The zeroth member of this class, having generating function f(s) = s, is the perfectly legitimate unfolding sequence (dots again as before):

7.2.6. S0 = 0.0.00.0000.00000000.0000000000000000.00000000000000000000000000000000...

The larger n, the more slowly the sequence grows (well, obvious, right?).

Can these Sn be mapped from V, the homomorphic mother? Of course they can--left as a challenge for the mildly interested reader.

It seems that my interest in these matters has been inextinguishable since the mid 1950s, from a short paper published under the title "Unending Chess and a Problem in Semigroups." The Wine Tasting Sequence is equivalent to the Hedlund-Morse symbolic trajectory (although the further thoughts about it and its family are mine); Prof. Hedlund of Yale was the author of the short paper; Profs. Hedlund, Morse, and Kakutani, mostly of Yale, are responsible for the earliest thoughts on the subject that I know of (Norwegian Axel Thue may earlier have steered attention in this general direction), and I bless the memories I have of them, hoping that they are all still with us, not to mention the Mathematics Library in Leet-Oliver Hall, HIllhouse Avenue, Yale University, New Haven, Connecticut. I was a kid. They were giants.

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