Orbit space of an irrational rotation

For irrational $\theta$ , let $T_\theta(x)=x+\theta$ on $\mathbb T$ . Its orbit space is $Q=\mathbb T/{\sim}$ , where $x\sim y$ means $y=x+n\theta$ for some $n\in\mathbb Z$ . Every orbit is countable and dense, while $Q$ has many distinct points. The quotient topology on $Q$ is indiscrete: its only open sets are $\varnothing$ and $Q$ .

Why functions collapse

A continuous function constant on each orbit is constant on the dense orbit of any one point and hence on the whole circle. A nonempty proper saturated open subset would have a nonempty invariant closed complement containing a dense orbit, a contradiction. This proves the quotient-topology assertion.

Haar ergodicity also gives $L^\infty(\mathbb T)^\alpha=\mathbb C1$ , even though invariant null sets such as individual orbits exist. The crossed product retains functions and implementing motion and provides a richer algebraic object associated with this quotient.

References

Alain Connes, Noncommutative Geometry, 1994, Introduction, examples of quotient spaces.