My question is about discrete dynamical systems, or simply, maps.
The $n$-dimentional diffeomorphism $\mathbf{F}: \mathbf{x}_n\mapsto \mathbf{x}_{n+1}$ is called the area-preserving map if the determinant of Jacobi matrix (i.e. Jacobian) of the map always equals to 1.
I know that area-preserving mapping can't have an attracter or repeller, but is there any area-preserving map that has chaos attractors? If possible, please give me an example.
Thank you.
Poking around on Google Scholar, it would appear that area-preserving maps can be chaotic. In particular, this paper, "Renormalization and Transition to Chaos in Area Preserving Nontwist Maps":
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.2069&rep=rep1&type=pdf
reports that:
$$y_{i+1} = y_i - b \sin (2 \pi x_i)$$ $$x_{i+1} = x_i + a (1-y_{i+1}^2)$$
has a chaotic region.