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## Backward and Forward with the Towel Map

Consider the family of symmetric towel maps (the towel terminology is due to S. Newhouse):

$f(x,y)=(1-ax^2-cy^2,1-ay^2-cx^2).$

This looks like a very natural generalization of quadratic interval maps, a step beyond the Viana maps of the form $f(x,y)=(dx \mod 1,1-ay^2+c\sin(2\pi x)$. These maps can also be understood as a coupling of two chaotic interval maps.

One would like to prove things like a two-dimensional version of Jakobson theorem. However little is known about these dynamics, except for their measures of maximal entropy which I was able to study using the entropy-expansion condition (for small enough $|c|$).

Now, let $a=1.8$, $c=0.2$ and iterate a random point of $[0,1]^2$ forward….

or backward (chosing randomly between the preimages) at each step:

Nice pictures, aren’t they? A towel and its diffraction pattern 😉

More on this later, hopefully…