# Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit cycles to limit cycles (and presumably preserves their arrangement in state space, if they are, for example, nested).

I have a question in the spirit of algebraic topology: Is there an interesting way to assign some algebraic structure to a continuous dynamical system so that equivalent systems have related structures associated with them (perhaps isomorphic)? I say "interesting" to exclude trivial things like assigning the same algebraic object to every system.

I am fairly sure I am not asking about "algebraic dynamics", which seems to be about discrete dynamical systems. I might be asking something related to "topological dynamics", but to be clear this question could perfectly well be about dynamical systems on e.g. $\mathbb{R}^2$, I don't think it is necessary to talk about general topological spaces.

• Are you familiar with symbolic dynamics? This is a way of encoding the essential dynamical features of a topological dynamical system using purely symbolic techniques. This may be related to what you're interested in, though admittedly your question is (purposely) vague. – Dan Rust Oct 20 '14 at 10:28
• You may also be interested in the spectrum of a dynamical system. – Dan Rust Oct 20 '14 at 10:53