An automorphism can be applied any integer number of times. Is there a sensible notion of applying it a non-integer number of times?
I have the same question for more general types of transformations.
EtA: I'm curious about the most general version of this question, but the motivating example is automorphisms of real or complex vector spaces.
EtA(x2-3): An example would be (for the reals): $f(x) = 2x$, $f^n(x) = 2^nx$. On the other hand, if $f(x) = -x$, then we can still define $f^n(x) = (-1)^nx$, but this can take complex values.