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Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of modulus 1. What is an easy way to prove that $\varphi$ is ergodic?

It's true that the stable and unstable manifolds at $(0,0) \in \mathbb{T}^2$ (projections of the eigenspaces of $A$ to the torus) are dense in the torus, which can be used to prove that such maps are topologically mixing. An example is Arnold's cat map.

Arnold and Avez show in Ergodic Problems in Classical Mechanics that Arnold's cat map is ergodic by proving that it has "Lebesgue spectrum", which implies that it is strong mixing, which implies that it is ergodic. Is there a more direct way to prove this?

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    A few weeks ago, someone here asked for a proof that these maps are chaotic. I found one in a textbook by Elaydi on discrete dynamical systems. The hardest part was proving the map transitive.2012-12-20
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    @GerryMyerson I saw [that thread](http://math.stackexchange.com/questions/248914/prove-that-anosov-automorphisms-are-chaotic). I found a simpler proof of a stronger statement (topologically mixing) in the [book by Broer and Takens](http://www.math.rug.nl/~broer/pdf/nova.pdf) (proposition 2.15, page 111). But I don't see how topological transitivity or mixing can help me prove that these maps are ergodic.2012-12-20

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