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**HallsofIvy** We have to be a little bit careful here. Because you are talking about linear mappings in the sense of "chaos theory", "periodic" refers to composition, not f(x+ p)= f(x) as in "periodic" trig functions.

Saying that a function, f, has "period 1" means that f(x)= x for all x.

Saying that it has "period 2" means that f(f(x))= x for all x.

For what a is f(x)= ax= x? If a is not that value, is there a value that makes [tex]f(f(x))= a(ax)= a^2x= x[tex]?

Suppose a not either of those values. Is it possible that can "periodic with period n"- that is that $\displaystyle f^n(x)= x$? Here, "$\displaystyle f^n(x)$ means the composition: f(f(f(...f(x)...))). And, for f(x)= ax, that is $\displaystyle a^nx$.