Determine which sets are local attractors and determine global attractors

Question

Consider the discrete-time dynamical system on $X=\mathbb R_0^+$ given by iteration of the map $f(x)=x^{1/2}$

I want to determine which of the sets $I_1=\{ 0 \}$, $I_2=\{ 1 \}$, $I_3=[0,1]$ are local attractors and if they are, determine their basin attractors.

$I_1$, $I_2$ and $I_3$ are compact invariant sets.

I then want to find the global attractor.

What I have worked out so far

So, first I have found the flow operator/map operator $S_{n}x$, $n \in \mathbb{N_0}$ to be

$S_{n}x=x^{\frac{1}{2}^n}$

and $\lim_{n \to \infty}S_nx = \lim_{n \to \infty} x^{\frac{1}{2}^n} = 1$

I also know the definition of a local attractor states that

A set $\Omega \subset X$ is called a local attractor if $\Omega$ is

i.) Compact

ii.) Invariant

iii.) There is a neighbourhood $V$ of $\Omega$ which is uniformly attracted by $\Omega$

Also, the Basin of Attraction of Omega is the set $A(\Omega)= \{ x \in X : dist(S_tx,\Omega) \rightarrow 0 \}$

We know $I_1$, $I_2$ and $I_3$ are all compact invariant sets so I just want to determine if there is a there is a neighbourhood $V$ of $\Omega$ which is uniformly attracted by $\Omega$ but I do not understand how to do this.

Similary I know that a global attractor is defined as follows:

A set $\Omega \subset \mathbb R^n$ is called a global attractor, if $\Omega$ is

• compact

• invariant

• attracts the bounded sets B ⊂ X

Again I am confused about how to prove $\Omega$ attracts the bounded sets B ⊂ X.

Any help much appreciated. Many thanks

Answer 1

You have shown that for all $x>0$, $\lim_{n\to\infty}S_nx=x^{2^{-n}}=1$. This tells you that $\{1\}$ is a local attractor. If $x=0$ we have $S_nx=0$ for all $n$. This implies that $\{1\}$ is not a global attractor since it does not attract any bounded set containing $0$.

Reasoning like this, you can see that $\{0\}$ is not a local attractor and that $[0,1]$ is a global attractor.