$Let (F, X) = $ dynamical system

Part 1: Prove that if $F(a) = a$ such that a $\epsilon$ X (a is a fixed point of F), then the basin of attraction of $a$ , $A_a$ is non-empty.

Def'n: Let (F, X) = dynamical system and let a $\epsilon$ X. The basin of attraction

of a is the set $A_a$ = {y $\epsilon$ X : $\lim_{a\to \infty} F^{n}y = a$}

Should I use proof by contradiction here?

Part 2: Fix b $\epsilon$ X. Prove that if R $\subseteq$ $A_{y}$(basin of attractn)then $F(R^{-1})$ $\subseteq$ $A_{y}$.

  • 1
    $\begingroup$ Could you please check against the original for typographical errors? The running index in the definition of $A_a$ is $n$, in part 2 I'm quite sure that $b=y$ and that it is not the inverse of the point set, but of the function. $\endgroup$ – Lutz Lehmann Jan 17 at 11:32
  • 1
    $\begingroup$ Please tell more of your own efforts and how far you got. Is part 1 trivially true as $a\in A_a$? $\endgroup$ – Lutz Lehmann Jan 17 at 11:34

Your Answer

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

Browse other questions tagged or ask your own question.