For example you have a $F(\cdot)$ which starts off taking some input $a$, which describes some set of objects and their states, and creating a output $F(a)$, which can be renamed $F_1$, that also describes a set of objects and their states, which is then input into $F$, to get $F_2 = F(F_1)$, or $F(F(a))$.

If this is repeated for an extensive period is it guaranteed that that the values of the individual iterations will be loop? For example $F(F_{x+n})=F(F_{x+2n})$ etc and $F(F_{x+m})=F(F_{x+2m})$ where $x$ is the start of the cycle and $m$ and $n$ are the frequency of the cycle.


As is usual with these sort of questions, the answer is "it depends".

For example, suppose $$ F(x) = x $$ then the loop is immediate.

For example, suppose $$ F(x) = x/2 $$ then the loop never occurs unless $x = 0$.

The same will be true for more complex $F$s.

For example $$ F(x) = \mathbf{A} x $$ where $\mathbf{A}$ is a matrix. If $\mathbf{A}$ is a rotation matrix of a certain type (so that it rotates by $360/n$ degrees where $n$ is an integer) then the period will be $n$. If $\mathbf{A}$ is something else, then all bets are off.

| improve this answer | |
  • $\begingroup$ this is kinda what eigenfunctions are about. $\endgroup$ – robert bristow-johnson Jul 13 '17 at 5:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.