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 $\psi_t$.

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

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.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.