# homeomorphism of flows wrt sets

Given two flows, $\phi_t: A \to A$, and $\psi_t:B \to B$, that are topologically conjugate, and a homeomorphism, $h: A\to B$, show the following relationships to be true. In the following, $x \in A$ and $y \in B$,

$$i.) \omega(y) = h(\omega(h^{-1}(y))) - \textrm{the omega limit set}\\ ii.) \text{if } \Lambda \textrm{ is invariant set of } \phi_t, \textrm{ i.e. } \phi_t(\Lambda) \subset \Lambda, \textrm{ then show } h(\Lambda) \textrm{ is invariant for } \psi_t \\ iii.) \textrm{if } W^S(\Lambda) \textrm{ is a basin of attraction of an attracting set } \Lambda, \textrm{ then so is } h(W^S(A)) \textrm{ for } h(\Lambda) \\ iv.) \textrm{ if } \Lambda \textrm{ is an attractor, then so is } h(\Lambda)$$

I know there is a commutative diagram $x \stackrel{\phi_t}{\to} \phi_t(x) \stackrel{h}{\to} \psi_t(y) \stackrel{\psi_t^{-1}}{\to} y \stackrel{h^{-1}}{\to} x$. Also, I know that fixed points and periodic orbits have their characteristics preserved by $h$.

The following two bold statements are from Meiss's book.

Periodic orbits: For instance, one can prove the claim about periodic orbits, $\psi_{t+T}(x_0) = \psi_t(x_0)$, by saying that $\psi_t(y_0) = h(\phi_t(x_0)) = h(\phi_{t+T})(x_0) = \psi_{t+T}(y_0)$, which effectively proves that for one point of the orbit, out of a continuum of them, this holds and therefore must hold for any arbitrary point in the periodic orbit.

Equilibria: Similarly, for $\phi_t(x^*) = x^*$, $\psi_t(h(x^*))=h(x^*) = y^*$, so $y^*$ is an equilibria for $\psi_t$.

• What is your question? And what did you try? – tomasz Dec 24 '13 at 0:38

## 1 Answer

The first exercise can be re-written as $\omega(y) = h(\omega(h^{-1}(y))) \;\longrightarrow\; \omega(y) = h(\omega(x)),$ which means that, for this to be proven true implies that the homeomorphism preserves the omega-limit sets.

For $\phi$;

$\omega(x) = \{x' \in A\; \large| \;\exists \; t_n \to \infty, \; \phi_{t_n} \to x' \}$ Let $y=h(x)$, then $\omega(y;\psi) = h(\omega(x;\phi))$.

So to start, let $y' \in \omega(y,\psi)$, then there exists $t_n \to \infty$ s.t. $\psi_{t_n}(y) \to y'$. Next, $h^{-1}(\psi_{t_n}(y))$ converges to a point $x' = h^{-1}(y') \in A$, i.e. $h^{-1}(\psi_{t_n}(y)) = \phi_{t_n}(x).$

As a result, $\phi_{t_n}(x) \to x'$ and so $x' \in \omega(x,\phi)$. Since $y' = h(x')$, one attains $y' \in h(\omega(x;\phi)),$ which shows that $\omega(y;\psi) \subset h(\omega(x;\phi)).$

Proving the inclusion the other way uses the same arguments, so that one has $\omega(x;\phi) \subset h^{-1}(\omega(y,\psi)),$ and therefore $h(\omega(x;\phi)) \subset \omega(y;\psi).$

As inclusion is proved both ways, one has the result that $\omega(y;\psi) = h(\omega(x;\phi)).$

By the properties of a homeomorphism, know that $\psi_t(h(A)) = h(\phi_t(A)),$ and restricting to the subset $\Lambda \subset A$, this becomes (still valid) the following, $\psi_t(h(\Lambda))=h(\phi_t(\Lambda)) \;\longrightarrow\; \psi_t(h(\Lambda)) \stackrel{\textrm{invariant}}{=} h(\Lambda) \;\longrightarrow\; \psi_t(h(\Lambda)) = h(\Lambda) \;\Rightarrow\; h(\Lambda)$ is invariant under $\psi_t$ in $B$.

\begin{equation*} \begin{array}{ccc} \psi_t(h(\Lambda))&=&h(\phi_t(\Lambda)) \\ \bigg\downarrow & & \downarrow \\ \psi_t(h(\Lambda)) & & h(\Lambda) \\ \end{array} \end{equation*}

which implies that $\psi_t(h(\Lambda)) = h(\Lambda),$ and that $h(\Lambda)$ is invariant under $\psi_t$ in $B$.

Using some abbreviations: (BOA is a basin of attraction, and AS is an attracting set.)

Know that $W^S(\Lambda)$ is a BOA of AS $\Lambda$, as well as knowing that $h(\Lambda)$ is an AS.

Consider the following. $h(\Lambda_A) = h\left( \cap_{t>0}\,\phi_t(N_A)\right) = \cap_{t>0}\,h\left(\phi_t(N_A)\right) = \cap_{t>0}\, \psi_t\left( h(N_A) \right) \longleftrightarrow \cap_{t>0}\,\psi_t(N_B) \;\Rightarrow\; h(N_A) = N_B$

In light of this, one can then proceed as follows. As $h(\Lambda)$ is an AS, it has a BOA, determined in the following. (Note that a trapping region [TR] is cpt. and $\psi_t(N_B) \subset \operatorname{int}\,N_B$ for $t>0$.) \begin{equation*} W^S(h(\Lambda)) = \cup_{t \leq 0}\,\psi_t(N_B) = \cup_{t\leq 0}\,\psi_t(h(N_A)) = \cup_{t \leq 0}\,h\left(\phi_t(N_A)\right) = h\,\left( \cup_{t \leq 0}\,\phi_t(N_A) \right) = h\left( W^S(\Lambda) \right) \end{equation*}

As $\Lambda$ is an attractor, it satisfies the two following properties.

 $\Lambda$ is an AS.

$\exists \;x \in A \;\textrm{ s.t. } \; \Lambda = \omega(x)$.


The attempted solution follows:

There exists a compact $N_A \supset A$ s.t. $\Lambda = \bigcap_{T>0}\,\phi_T(N_A)$. Since $W^S(\Lambda)$ is a (maximal) trapping region, let $\Lambda = \bigcap_{T>0}\,\phi_T\left( \bigcup_{t\leq 0}\,\phi_t(N_A)\right) = \bigcap_{T>0}\,\phi_T\left(W^S(\Lambda)\right) = \bigcap_{T>0}\,\left(h^{-1} \circ \psi_T \circ h\right)\,(W^S(\Lambda)) \longrightarrow \bigcap_{T>0}\,\left( h^{-1} \circ \psi_T \right)\,W^S(h(\Lambda)) \stackrel{\textrm{cts}}{=} h^{-1}\,\bigcap_{T>0}\,\psi\left( W^S(h(\Lambda))\right) \;\longrightarrow\; h(\Lambda) = \bigcap_{T>0}\,\psi\left(W^S(h(\Lambda))\right)$ which is again a trapping region (maximal) and warrants claiming that $h(\Lambda)$ is an AS.

Consider the following succession. $\omega\left(h^{-1}(y)\right) = h^{-1}(\omega(y)) \longrightarrow h(\Lambda) = h\left(\omega(x)\right) =\omega(y) \;\Rightarrow\; \exists\; y \textrm{ s.t. } h(\Lambda) = \omega(y)$

Therefore, $h(\Lambda)$ is the attractor associated to $\Lambda$ via the homeomorphism.