# Questions regarding dynamics

$$Let (F, X) =$$ 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 $$F(R^{-1})$$ $$\subseteq$$ $$A_{y}$$.

• 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. – Lutz Lehmann Jan 17 at 11:32
• Please tell more of your own efforts and how far you got. Is part 1 trivially true as $a\in A_a$? – Lutz Lehmann Jan 17 at 11:34