More on Leroy Quet's permutation transform

[David Wasserman]

To Leroy Quet: Do you think it's appropriate for me to refer to it as the "Quet transform"?  Since no one has posted a reference, I'm assuming it's original.  However, maybe you have invented (or will invent) some other transform that you'd rather give your name to.

v/r,
David
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The transform of a periodic sequence is not necessarily periodic.  For example, the sequence 1,2,1,2,1,2... transforms to itself, but 1,2,3,1,2,3... transforms to a sequence that begins 1,3, and then repeats (1,4,1).  Repeat(1,2,3,4) transforms to 1,4,Repeat(1,6,1,2).  Repeat(1,2,3,4,5) transforms to 1,5,1,8,Repeat(1,3,1,9,1).  

I'm sure that the transform of a periodic sequence will always be eventually periodic, but I'm not sure that it will always become periodic within the length of one cycle, as these examples do.

More generally, the transform of an eventually periodic sequence will always be eventually periodic.  For example, 1,3,1,Repeat(4,1,1) transforms to Repeat(1,2,3) (the Quet transform is its own inverse), while Repeat(4,1,1) transforms to 2,2,3,Repeat(1,2,3).  Thus when I changed the nonrepeating part of the sequence, the change in the transformed sequence was also confined to the nonrepeating part.  Is this always true?  More generally, given two sequences (eventually  periodic or not) that differ in finitely many places, can their transforms differ in infinitely many places?