Are positive Lyapunov exponents and strange attractors equivalent conditions for chaos?

[mersecske]

Usually positive Ljapunov exponent is said to be a condition for chaos,
and the strange atractor is so.

Are there any theorem which shows that the two condition is equivalent?
They are necessary and sufficent conditions?

 

[Jack21222]

The necessary conditions for chaos are topological mixing and sensitive dependence of initial conditions.

 

[mersecske]

Where can I find a precise statement and proof of "non-invertibility is necessary condition for chaos in 1D" ?