Dynamical Systems help

Question

Can anyone help me with this please?

Consider the finite set $ S = \{a, b, c, d\}$

Suppose we know the function $f : S \rightarrow S $ has the property that

$f(a) = b, f(c) = d, f(b) $is not equal to $ b\ , f(d)$ is not equal to $ d$.

Prove that the dynamical system $(S, f)$ does not have a fixed point.

What did you try? And what does mean "f(b) 6=b" ? Maybe it's $f(b)\neq b$? — paf, Aug 12 '16 at 10:38
Yes sorry i dont know where the 6 came from!! I'm not quite sure what to try because i dont understand it — Karensav93, Aug 12 '16 at 10:40
Do you understand what is a fixed point? If yes, it suffices to check that none of the 4 points of S is a fixed point! — paf, Aug 12 '16 at 10:45
Yes i understand what a fixed point is and usually to find a fixed point you let the equation equal to x, but there isn't any given equation here so how do i check that none of the 4 points are fixed points? — Karensav93, Aug 12 '16 at 10:48
I put the answer as community wiki just so that the question gets removed from the unanswered queue. — Dan Rust, Aug 12 '16 at 12:11

score 1 · Answer 1 · answered Aug 12 '16 at 12:11

1

$f(a)=b \neq a$ so $a$ is not fixed.

$f(b) \neq b$ so $b$ is not fixed.

$f(c)=d \neq c$ so $c$ is not fixed.

$f(d)\neq d$ so $d$ is not fixed.

As all four points of $S$ are not fixed by $f$, the dynamical system $(S,f)$ has no fixed points.

answered Aug 12 '16 at 12:11

Dan Rust

1 Answers1