Cat map like maps without period

Question

Is there any area-preserving chaotic map other than Arnold cat map which can be applied on a rectangle as well as being reversible but not periodic?

Answer 1

Sure. There are Cat maps other than Arnold's Cat map. One general form of 2D Cat map is $${\bf{C}} = \begin{bmatrix}1 & a \\b & ab+1\end{bmatrix}$$

Answer 2

Irrational rotations of the torus provide such examples. Even on the 1D torus, or circle, it's easy to see that an irrational rotation has no fixed point.

Also, when viewed as a map on the rectangle, the cat map is not continuous, so Brouwer's theorem does not apply. More properly, the cat map can be viewed as a map on the torus where it is continuous. The fact that it has all kinds of periodic points does not follow from Brouwer's theorem, however.