What is Topological Equivalence in Functions and Dynamical Systems?

[marellasunny]

It would be helpful if someone could please explain topological equivalence of functions in simple words?
I am working on dynamical systems and chaos theory.In the underlying material,topological equivalence has taken a more complex definition involving orbits.Please be kind enough to explain this part also.
'What does it mean for 2 maps to be topologically equivalent'?
Thankyou.

 

[K41]

The dynamics of one system A are topologically equivalent to the dynamics of another system B, if a homeomorphic function exists such that it can map from one vector A to the other vector B and preserve the direction of time.

 

[marellasunny]

Is this the same definition of 'holomorphic function' we use in complex analysis?i.e A function f is holomorphic at a point z if it is C differentiable in a neighborhood of z/preserves angles and orientation in space.??Does preserving the direction of time also come into special consideration only for dynamical systems?

Are the topological properties of a dynamical system the same as the properties of a topological space?

What does it mean when one says the perturbed system is topologically isomorphic to the unpurturbed system?

 

[HallsofIvy]

No one has mentioned "holomorphic". djpailo used the word "homeomorphic".

A "homeomorphism" is an invertible, continuous, function from one topological space to another and two topological spaces are "equivalent" if and only if there exist a homeomophism from one to the other.