Topological conjugacy, and related-but-distinct § Topological equivalence of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.
are iterated functions, and there exists a homeomorphism
Then one must have and so the iterated systems are topologically conjugate as well.
denotes function composition.
are continuous functions on topological spaces,
is furthermore injective, then bijective, and its inverse is continuous too; i.e.
[2] Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring
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.
are mapped to homeomorphic orbits of
makes this fact evident:
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
, which is requiring more than simply that orbits of
This motivates the definition of topological equivalence, which also partitions the set of all flows in
into classes of flows sharing the same dynamics, again from the topological viewpoint.
homeomorphically, and preserving orientation of the orbits.
In addition, one must line up the flow of time: for each
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 to 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.
More equivalence criteria can be studied if the flows,
Two dynamical systems defined by the differential equations,
Two dynamical systems on the same state space, defined by
, are said to be orbitally equivalent if there is a positive function,
Orbitally equivalent system differ only in the time parametrization.
For example, consider linear systems in two dimensions of the form
, 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 orbitally equivalent or smoothly equivalent,[5] because their eigenvalues are different (notice that the Jacobians of two locally smoothly equivalent systems must be similar, so their eigenvalues, as well as algebraic and geometric multiplicities, must be equal).
There are two reported extensions of the concept of dynamic topological conjugacy: This article incorporates material from topological conjugation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.