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Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} \end{align}

I already proved that $f$ is invariant with respect to the two dimensional Lebesgue measure on $[0,1]\times[0,1]$. Now I'm trying to prove that $f$ is ergodic with respect to the same measure, but I couldn't formalize any argument. I picked some squares $A,B$ on the plane and tried to calculate $\lambda(f^{-n}A\cap B)$ to prove that $f$ is in fact mixing, the draws went pretty well, but again, couldn't formalize the arguments. I tried some other equivalent forms of ergodicity, but none seems to help me. Any suggestion?

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Isomorphisms preserve ergodicity. So if you can construct an isomorphism of the baker map to a bernoulli shift you are done. I think this is the standard way it is done in most books.

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    $\begingroup$ How could this be done without showing that it is isomorphic to bernoulli shift. $\endgroup$ – Neil hawking Dec 25 '18 at 12:07

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