Define two dynamical systems $([0,1), \mathbb{B}\_1, \mathbb{L}, T\_1(x)=x+\alpha\_1\mod 1)$ and $([0,1), \mathbb{B}\_2, \mathbb{L}, T\_2(x)=x+\alpha\_2\mod 1)$, where $\alpha\_1,\alpha\_2 $ are two irrational numbers, $\mathbb{L}$ is the Lebesgue measure and $\mathbb{B}\_i$ are the sigma algebras. Now if these two systems are isomorphic, then what the relationship between $\alpha\_1$ and $\alpha\_2 $? I do know that the irrational rotation is isomorphic to the Sturmian orbit dynamical systems. 



Isomophism between two irrational rotations
The first major classification result in ergodic theory was found by Halmos and von Neumann in 1942 (*Operator methods in classical mechanics II*, Ann. of  Math. (2) 43 (1942), 332–350.). They showed that for ergodic rotations of compact abelian groups, spacial isomorphism (i.e. measure-theoretic isomorphism) is the same as spectral isomorphism (i.e. unitary equivalence of the associated operators). For irrational rotations $\alpha\_1$ and $\alpha\_2$ of the circle, this amounts saying that the subgroups generated by $\alpha\_1$ and $\alpha\_2$ are equal. From this it is easy to deduce that $\alpha\_1 = \pm \alpha\_2$. This is simply an expanded version of Anthony's initial answer.


