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 formula_1 and formula_2 are iterated functions, and there exists a homeomorphism formula_3 such that 
so that formula_1 and formula_2 are topologically conjugate. Then one must have 
and so the iterated systems are topologically conjugate as well. Here, formula_8 denotes function composition.
Definition.
formula_9, and formula_10 are continuous functions on topological spaces, formula_11 and formula_12.
formula_1 being topologically semiconjugate to formula_2 means, by definition, that formula_3 is a surjection such that formula_16.
formula_1 and formula_2 being topologically conjugate means, by definition, that they are "topologically semiconjugate" and formula_3 is furthermore injective, then bijective, and its inverse is continuous too; i.e. formula_3 is a homeomorphism; further, formula_3 is termed a topological conjugation between formula_1 and formula_2.
Flows.
Similarly, formula_24 on formula_25, and formula_26 on formula_27 are flows, with formula_28, and formula_29 as above.
formula_24 being "topologically semiconjugate" to formula_26 means, by definition, that formula_3 is a surjection such that formula_33, for each formula_34, formula_35.
formula_24 and formula_26 being "topologically conjugate" means, by definition, that they are "topologically semiconjugate" and is a homeomorphism.
Discussion.
Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring formula_1 and formula_2 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 formula_2 are mapped to homeomorphic orbits of formula_1 through the conjugation. Writing formula_42 makes this fact evident: formula_43. 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 formula_44 and formula_45 to be topologically conjugate for each formula_46, which is requiring more than simply that orbits of formula_24 be mapped to orbits of formula_26 homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in formula_25 into classes of flows sharing the same dynamics, again from the topological viewpoint.
Topological equivalence.
We say that two flows formula_24 and formula_26 are topologically equivalent, if there is a homeomorphism formula_52, mapping orbits of formula_26 to orbits of formula_24 homeomorphically, and preserving orientation of the orbits. In other words, letting formula_55 denote an orbit, one has
for each formula_57. In addition, one must line up the flow of time: for each formula_57, there exists a formula_59 such that, if formula_60, and if is such that formula_61, then formula_62.
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.
Smooth and orbital equivalence.
More equivalence criteria can be studied if the flows, formula_24 and formula_26, arise from differential equations.
Two dynamical systems defined by the differential equations, formula_65 and formula_66, are said to be "smoothly equivalent" if there is a diffeomorphism, formula_67, such that
In that case, the dynamical systems can be transformed into each other by the coordinate transformation, formula_69.
Two dynamical systems on the same state space, defined by formula_70 and formula_71, are said to be "orbitally equivalent" if there is a positive function, formula_72, such that formula_73. 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 formula_74. If the matrix, formula_75, 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, 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).
Generalizations of dynamic topological conjugacy.
There are two reported extensions of the concept of dynamic topological conjugacy: