Thanks for asking
"In what case?" Good question. At the risk of boring you, here's the context
(which I recommend just skimming) --
Arnie next explained to me that most of the previous
results were but special cases of one very lovely general
fact involving the important concept of Fixed Points.
I shall use the symbol M to denote any expression having
the property that for every expression X, the expression
M2X yields some expression Y, and for any Z other than
2X that yields X, MZ yields the same Y as is yielded by
M2X. Such an expression M, I will call a Special Expression.
Thus for any Special Expression M, if Y1 and Y2 yield the
same expression, then MY1 and MY2 will yield the same
expression. Any expression composed of C, R, and V is
special. For any Special Expression M, and any
expression X, by M(X) I shall mean the expression yielded
by M2X (or by MY for any Y that yields X). For example
R(X) is the repeat of X (since R2X yields XX) and V(XYZ)
is the reverse of XYZ (since V2XYZ yields ZYX).
Is this clear so far?
If I was in your shoes only that next to last sentence might be clear and
that's all that needs to be clear when the source then states --
Now I call X a Fixed Point of [function] M if X yields M(X).
Many of the previous problems were tantamount to finding
Fixed Points. For example, finding an X that yields its
own repeat is finding a Fixed Point of [M = ] R; an X that
yields its own reverse is simply a Fixed Point of [M = ] V;
and so forth.
Let me add another example of all of the above to
clarify the meaning of "X yields" --
For M = R, the Repeat function, if I choose the string (and
function) X to equal R32R3 then X yields R32R3R32R3 and
M(X) = R(R32R3) = R32R3R32R3, so X = R32R3 is a
of function M = R in this case.
And it is my interpretation that X is a string and a "function"
(of itself!). You can interpret X as the concatenation of
right-to-left Operators and an Operand, with the two
separated by the first '2' you come to from the left.
The discovery of what X yields M(X) is not at all obvious at first.
X = R32R3 as a string/function is evaluated, from right to left, as --
2R3 = R3 then 3(R3) = R32R3 then R(R32R3) = R32R3R32R3.
Whereas M(X) = R(X) = R(R32R3) = R32R3R32R3 is a
simple, straightforward repeat of X.
As you may observe, 2X simply yields X and 3(X) yields
X2X (and 2X is simply the string concatenation of 2 and X,
not 2 times X; and, e.g. 3(X) does not equal 3X ).
Getting back to the Wikipedia definition of a Fixed Point
where f(c) = c, I can think of no X where M(X) = X but
the source calls X a Fixed Point
of [function] M if
X yields M(X) or, if M(X) equals what X yields
NOT if M(X) equals X. That's my problem...
EDIT: Well I guess using the Reverse function V --
V(X) does equal X for X as any palindrome but
by the source's definition the Fixed Point
the function V(X) occurs at X = V32V3...