# Topological conjugacy

In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterated function can be solved, then those for any topologically conjugate function follow trivially.

To illustrate this directly: suppose that Template:Mvar and Template:Mvar are iterated functions, and there exists an Template:Mvar such that

${\displaystyle g=h^{-1}\circ f\circ h,}$

so that Template:Mvar and Template:Mvar are topologically conjugate. Then of course one must have

${\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h,}$

and so the iterated systems are conjugate as well. Here, ○ denotes function composition.

## Definition

Let Template:Mvar and Template:Mvar be topological spaces, and let ${\displaystyle f\colon X\to X}$ and ${\displaystyle g\colon Y\to Y}$ be continuous functions. We say that Template:Mvar is topologically semiconjugate to Template:Mvar if there exists a continuous surjection ${\displaystyle h\colon Y\to X}$ such that ${\displaystyle f\circ h=h\circ g}$.

If Template:Mvar is a homeomorphism, we say that Template:Mvar and Template:Mvar are topologically conjugate and we call Template:Mvar a topological conjugation between Template:Mvar and Template:Mvar.

Similarly, a flow Template:Mvar on Template:Mvar is topologically semiconjugate to a flow Template:Mvar on Template:Mvar if there is a continuous surjection ${\displaystyle h\colon Y\to X}$ such that ${\displaystyle \varphi (h(y),t)=h\psi (y,t)}$ for each ${\displaystyle y\in Y}$, ${\displaystyle t\in \mathbb {R} }$. If Template:Mvar is a homeomorphism, then Template:Mvar and Template:Mvar are topologically conjugate.

## Examples

• the logistic map and the tent map are topologically conjugate.[1]
• the logistic map of unit height and the Bernoulli map are topologically conjugate. {{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B=

{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

## Discussion

Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring Template:Mvar and Template:Mvar to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of dynamical systems, since each class contains all functions which share the same dynamics from the topological viewpoint. For example, orbits of Template:Mvar are mapped to homeomorphic orbits of Template:Mvar through the conjugation. Writing ${\displaystyle g=h^{-1}\circ f\circ h}$ makes this fact evident: ${\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h}$. 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 ${\displaystyle \varphi (\cdot ,t)}$ and ${\displaystyle \psi (\cdot ,t)}$ to be topologically conjugate for each Template:Mvar, which is requiring more than simply that orbits of Template:Mvar be mapped to orbits of Template:Mvar homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in Template:Mvar into classes of flows sharing the same dynamics, again from the topological viewpoint.

## Topological equivalence

We say that two flows Template:Mvar and Template:Mvar are topologically equivalent, if there is a homeomorphism ${\displaystyle h:Y\to X}$, mapping orbits of Template:Mvar to orbits of Template:Mvar homeomorphically, and preserving orientation of the orbits. In other words, letting ${\displaystyle {\mathcal {O}}}$ denote an orbit, one has

${\displaystyle h({\mathcal {O}}(y,\psi ))=\{h(\psi (y,t)):t\in \mathbb {R} \}=\{\varphi (h(y),t):t\in \mathbb {R} \}={\mathcal {O}}(h(y),\varphi )}$

for each ${\displaystyle y\in Y}$. In addition, one must line up the flow of time: for each ${\displaystyle y\in Y}$, there exists a ${\displaystyle \delta >0}$ such that, if ${\displaystyle 0<\vert s\vert , and if Template:Mvar is such that ${\displaystyle \varphi (h(y),s)=h(\psi (y,t))}$, then ${\displaystyle s>0}$.

Overall, topological equivalence is a weaker equivalence criterion than topological conjugacy, as it does not require that the time term is mapped along with the orbits and their orientation. An example of a topologically equivalent but not topologically conjugate system would be the non-hyperbolic class of two dimensional systems of differential equations that have closed orbits. While the orbits can be transformed each other to overlap in the spatial sense, the periods of such systems cannot be analogously matched, thus failing to satisfy the topological conjugacy criterion while satisfying the topological equivalence criterion.

## Smooth and orbital equivalence

More equivalence criteria can be studied if the flows Template:Mvar and Template:Mvar arise from differential equations.

Two dynamical systems defined by the differential equations ${\displaystyle x'=f(x)}$ and ${\displaystyle y'=g(y)}$ are said to be smoothly equivalent if there is a diffeomorphism ${\displaystyle h:X\to Y}$ such that

${\displaystyle f(x)=M^{-1}(x)g(h(x))\quad {\text{where}}\quad M(x)={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}.}$

In that case, the dynamical systems can be transformed into each other by the coordinate transformation ${\displaystyle y=h(x)}$.

Two dynamical systems on the same state space, defined by ${\displaystyle x'=f(x)}$ and ${\displaystyle x'=g(x)}$, are said to be orbitally equivalent if there is a positive function ${\displaystyle \mu :X\to \mathbf {R} }$ such that ${\displaystyle g(x)=\mu (x)f(x)}$. Orbitally equivalent system differ only in the time parametrization.

Systems that are smoothly equivalent or orbitally equivalent are also topologically equivalent. However, the reverse is not true. For example, consider linear systems in two dimensions of the form ${\displaystyle x'=Ax}$. If the matrix ${\displaystyle A}$ has two positive real eigenvalues, the system has an unstable node; if the matrix has two complex eigenvalues with positive real part, the system has an unstable focus (or spiral). Nodes and foci are topologically equivalent but not smoothly or orbitally equivalent.[2]

## Generalizations of dynamic topological conjugacy

There are two reported extensions of the concept of dynamic topological conjugacy:

1. Analogous systems defined as isomorphic dynamical systems
2. Adjoint dynamical systems defined via adjoint functors and natural equivalences in categorical dynamics.[3][4]