Arnold's "cat map" is now one of the typical first examples of an Anosov diffeomorphism of the two torus $\mathbb{T}^2$. It is obtained by considering the linear map $A: \mathbb{R}^2 \rightarrow \mathbb{R^2}$ with matrix
\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}
with respect the the standard basis. Since $A$ preserves the integer lattice and has determinant one, it descends to an invertible map $f_A: \mathbb{T}^2\rightarrow \mathbb{T}^2$.
I believe I once read that the original cat image was due to Arnold, and that whatever the original source is hard to find in print (I cannot recall a source, so this could be completely wrong.) I have seen a variety of cats used as a substitute. Here is a possible first cat that I found on Google images. I recall at one point seeing a similar image in an old dynamics text (again, I can't recall the book), which looked similar to think linked image but was not solidly black, it was just an outline.
I would appreciate really appreciate any references or links!
I don't know if it's the first cat, but there is one on p. 6 in the 1968 book by Arnold & Avez, Ergodic Problems of Classical Mechanics.