Assume $f(x)$ is a 1D map exhibiting a period-3 orbit P. Then P is a period 3 orbit of $f(f(x))$?
I know this is a somewhat elementary question but I'm only asking it to confirm if my thought process is correct and applicable for more complex problems.
A period 3 orbit would mean $f(x) = (a \Rightarrow b \Rightarrow c \Rightarrow a \Rightarrow b \Rightarrow c........ etc)$
so would I be correct in saying $f(f(x)) = (b \Rightarrow c \Rightarrow a \Rightarrow b \Rightarrow c \Rightarrow a........ etc)$
and therefore P is the same period 3 orbit and the statement in my question is true. Would appreciate if anybody could verify that I am thinking about this in the correct way.
Thanks