Problem 62

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Covering space for $\Sigma_A \to \mathbb{T}^2$ corresponding to $\mathbb{R}^2 \to \mathbb{T}^2$



don't see what that means...

Steve Hurder's picture

Maybe $\Sigma_A$ means a 2-dimensional solenoid, and then he is asking about generalizations of the Smale attractor?

Often $\Sigma_A$ refers to a shift of finite type defined by a matrix $A$. Perhaps then  $\pi: \Sigma_A \to \mathbb T^2$ is a factor map from a mixing shift of finite type onto a hyperbolic toral automorphism derived from some Markov partition. Then he would be asking for something analogous to the universal cover $p: \mathbb{R}^2 \to \mathbb T^2$. A natural candidate would be the fiber product of $\pi$ and $p$, perhaps restricted to a suitable subsystem. But, for what problem is this useful ... 

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