# Can a CL-term have multiple fixed points?

Given a CL term $$E$$, can there exist multiple non-equivalent fixed points for $$E$$?

I think: any fixed point of $$E$$ can be expressed as $$Y(E)$$, this expression cannot reduce to multiple non-equivalent forms, due to confluence. So, I think that any CL term can have only 1 fixed point.

Is my logic correct? All ideas regarding this problem are welcome.

Edit:

Realized just now, the identity combinator is a CL term with multiple non-equivalent fixed points. But then, if we consider the term $$Y(I)$$, this must have multiple non-equivalent forms, corresponding to the multiple fixed points of $$I$$. But how is that justifiable under confluence?

Edit 2:

$$Y(I)$$ is nothing but the omega combinator. I think I have got it now. Even when a CL term has multiple fixed points, applying that CL term to Y gives us only one possible fixed point. Is that correct?

This may be useful to construct (partial) answers to your question: In general, there are infinitely many non-β-equivalent fixed-point combinators in λ-calculus. Consider $$\delta := λy.λx.x(yx) =_β \mathsf{SI},$$ then the terms $$Y_n$$ defined by $$Y_0 := Y \qquad Y_{n+1} := Y_n \delta$$ are fixed points combinators, and $$Y_i \neq_β Y_j$$.