| Version 4 |
Version 3 |
| Let $X$ and $Y$ be topological spaces, and let $f\colon X\to X$ and $g\colon Y\to Y$ |
Let $X$ and $Y$ be topological spaces, and let $f\colon X\to X$ and $g\colon Y\to Y$ |
| be continuous functions. We say that $f$ is |
be continuous functions. We say that $f$ is |
| \emph{topologically semicongugate} to $g$, if there exists a continuous |
\emph{topologically semicongugate} to $g$, if there exists a continuous |
| surjection $h\colon Y\to X$ such that $fh=hg$. If $h$ is a homeomorphism, |
surjection $h\colon Y\to X$ such that $fh=hg$. If $h$ is a homeomorphism, |
| then we say that $f$ and $g$ are \emph{topologically conjugate}, and we call |
then we say that $f$ and $g$ are \emph{topologically conjugate}, and we call |
| $h$ a \emph{topological conjugation} between $f$ and $g$. |
$h$ a \emph{topological conjugation} between $f$ and $g$. |
| Similarly, a flow $\varphi$ on $X$ is topologically semiconjugate to a flow $\psi$ on $Y$ if there is a continuous surjection $h\colon Y\to X$ such that |
Similarly, a flow $\varphi$ on $X$ is topologically semiconjugate to a flow $\psi$ on $Y$ if there is a continuous surjection $h\colon Y\to X$ such that |
| $\varphi(h(y),t) = h\psi(y,t)$ for each $y\in Y$, $t\in \R$. If $h$ is a homeomorphism then $\psi$ and $\varphi$ are topologically conjugate. |
$\varphi(h(y),t) = h\psi(y,t)$ for each $y\in Y$, $t\in \R$. If $h$ is a homeomorphism then $\psi$ and $\varphi$ are topologically conjugate. |
| \subsection{Remarks} |
\subsection{Remarks} |
| Topological conjugation defines an equivalence relation in the |
Topological conjugation defines an equivalence relation in the |
| space of all continuous surjections of a topological space to itself, |
space of all continuous surjections of a topological space to itself, |
| by declaring $f$ and $g$ to be related if they are topologically |
by declaring $f$ and $g$ to be related if they are topologically |
| conjugate. This equivalence relation is very useful in the theory of |
conjugate. This equivalence relation is very useful in the theory of |
| dynamical systems, since each class contains all functions which |
dynamical systems, since each class contains all functions which |
| share the same dynamics from the topological viewpoint. In fact, orbits |
share the same dynamics from the topological viewpoint. In fact, orbits |
| of $g$ are mapped to homeomorphic orbits of $f$ through the conjugation. |
of $g$ are mapped to homeomorphic orbits of $f$ through the conjugation. |
| Writting $g = h^{-1}fh$ makes this fact evident: $g^n = h^{-1}f^nh$. |
Writting $g = h^{-1}fh$ makes this fact evident: $g^n = h^{-1}f^nh$. |
| Speaking informally, topological conjugation is a ``change of coordinates'' in the topological sense. |
Speaking informally, topological conjugation is a ``change of coordinates'' in the topological sense. |
| However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps $\varphi(\cdot,t)$ and $\psi(\cdot,t)$ to be topologically conjugate for each $t$, which is requiring more than simply that orbits of $\varphi$ be mapped to orbits of $\psi$ homeomorphically. |
However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps $\varphi(\cdot,t)$ and $\psi(\cdot,t)$ to be topologically conjugate for each $t$, which is requiring more than simply that orbits of $\varphi$ be mapped to orbits of $\psi$ homeomorphically. |
| This motivates the definition of \emph{topological equivalence}, which also partitions the set of all flows in $X$ into classes of flows sharing the same dynamics, again from the topological viewpoint. |
This motivates the definition of \emph{topological equivalence}, which also partitions the set of all flows in $X$ into classes of flows sharing the same dynamics, again from the topological viewpoint. |
| We say that $\psi$ and $\varphi$ are \emph{topologically equivalent}, if there is an homeomorphism $h:Y\to X$, mapping orbits of $\psi$ to orbits of $\varphi$ homeomorphically, and preserving orientation of the orbits. This means that: |
We say that $\psi$ and $\varphi$ are \emph{topologically equivalent}, if there is an homeomorphism $h:Y\to X$, mapping orbits of $\psi$ to orbits of $\varphi$ homeomorphically, and preserving orientation of the orbits. This means that: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\mathcal{O}(y,\psi) = \{\psi(y,t): t\in\R\} = \{\varphi(h(y),t):t\in\R\}= \mathcal{O}(h(y),\varphi)$ for each $y\in Y$; |
\item $\mathcal{O}(y,\psi) = \{\psi(y,t): t\in\R\} = \{\varphi(h(y),t):t\in\R\}= \mathcal{O}(h(y),\varphi)$ for each $y\in Y$; |
| \item for each $y\in Y$, there is $\delta>0$ such that, if $0<|s|< t < \delta$, and if $s$ is such that $\varphi(h(y),s) = \psi(y,t)$, then $s>0$. |
\item for each $y\in Y$, there is $\delta>0$ such that, if $0<|s|< t < \delta$, and if $s$ is such that $\varphi(h(y),s) = \psi(y,t)$, then $s>0$. |
| \end{enumerate} |
\end{enumerate} |
