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We define the Smale set as $\Lambda=K \times K $ ($K$ is the Cantor set). $\{0,2\}^{\mathbb{Z}}$ is the set of functions from the integers to $\{0,2\}$. This theorem is linked to chaos theory, and to Smale's Horseshoe map (link here to wikipedia https://en.wikipedia.org/wiki/Horseshoe_map). The topology on $\{0,2\}$ is defined with neighborhoods, such that a nbhd of $\{a_k\}$ contains all the sequences $\{b_k\}$ such that $b_k=a_k$ for $|k| < n$ for a fixed natural $n$.

The function I am trying to prove is a homeomorphism sends a point of $\Lambda$ to its "history". Referring to wikipedia, let's call $A_0$ and $A_2$ the first two green horizontal lines here: https://upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Foldings2.png/220px-Foldings2.png

Now, if a point $x \in \Lambda$ is in $A_0$ (let's say the upper one), then $a_0=0$. Then I consider its image by the horseshoe map. If that is in $A_0$ then $a_1=0$, if it's in $A_2$ then $a_1=2$, and so on. For negative integers we use the inverse of the map. So I would like to prove that the application $\Phi:x \to \{a_k\}$ is an homeomorphism, but in fact I wouldn't know how to prove that is continuous, so...

P.S. To be really rigorous here would take forever, so I lacked a bit in formalism, but I think this is a quite famous result, so many people already know what this is all about

tommy1996q
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  • You can proceed for example as in https://math.stackexchange.com/questions/69905/the-cantor-set-is-homeomorphic-to-infinite-product-of-0-1-with-itself-cy?rq=1 (it is a different question but the procedure is the same). – John B Jan 22 '18 at 23:34
  • It's worth noting that the Smale set is homeomorphic to the Cantor set! – Alex Kruckman Jan 23 '18 at 15:20
  • Both sets are compact second countable Hausdorff spaces without isolated points. So they are homeomorphic, by Brouwer's characterisation of the Cantor set. – Henno Brandsma Jan 23 '18 at 17:38
  • Thanks, I think I have understood now – tommy1996q Jan 27 '18 at 17:58

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