Let and be topological spaces, and let and be continuous functions. We say that is topologically semiconjugate to , if there exists a continuous surjection such that . If is a homeomorphism, then we say that and are topologically conjugate, and we call a topological conjugation between and .

Similarly, a flow on is topologically semiconjugate to a flow on if there is a continuous surjection such that for each , . If is a homeomorphism then and are topologically conjugate.

Remarks

Topological conjugation defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring and 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. In fact, orbits of are mapped to homeomorphic orbits of through the conjugation. Writing 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 and to be topologically conjugate for each , which is requiring more than simply that orbits of be mapped to orbits of homeomorphically. 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.

We say that and are topologically equivalent, if there is an homeomorphism , mapping orbits of to orbits of homeomorphically, and preserving orientation of the orbits. This means that:

1. for each ;
2. for each , there is such that, if , and if is such that , then .