# Cat map like maps without period

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?

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What do you mean by the term "not periodic" ? Do you mean it should have no periodic points ? If so, then it is not possible because a continuous map on a compact manifold (rectangle) will have atleast one fixed point (which is just a periodic point of period 1) by Brouwer's theorem. –  nonlinearism Mar 14 '13 at 18:56

## 2 Answers

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.

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

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But this is still periodic. –  user39576 Mar 14 '13 at 18:16