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Assume a dynamical system is given on $S^1 = \mathbb{R} / \mathbb{Z}$ by $T(x)=x+\alpha$ mod $1$ where $\alpha$ is irrational. If we transform it into a symbolic system, for example, by setting $a_n=0$ iff $T^nx_0\in [0,0.5)$, would it be a shift of finite type?

Thanks in advance.

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    Please include definition of shift of finite type in the question. – 2017-01-14
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    Clearly not, independently of $\alpha$ and $x_0$. Also better "topological Markov chain". – 2017-01-14
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    Thanks. Is there any easy way to see that it's not an SFT? – 2017-01-15
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    Sure, you never get all symbols and that's immediate (just think of the orbits). If you prefer by brute force, it would be a topological factor gaining topological entropy. – 2017-01-15

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