If we expand modal logic with least and greatest fix point operators $\mu$ and $\nu$, respectively, we obtain the logic $L_\mu$.
An alternating automaton on infinite trees has a state space that is divided between two players, $\exists$ and $\forall$. Transitions of this automaton assign both a new state and a movement directive in the graph. Whether such an automaton accepts a tree is based on a two-player parity game. A player has to first choose a transition and depending on whether that transition involves a forward move or not, the game either remains in the same vertex or the player has to additionally choose one of the successors of the current vertex. If player $\exists$ wins this game with a positional strategy $s$ then the tree is accepted by the automaton.
It is now possible to translate such $L_\mu$ formulae to alternating automata on infinite trees by introducing a state for each subformula and choosing the transition function in a way that reflects the current formula. This would mean if the current formula is a disjunction of two subformulae, player $\exists$ could choose between the two states representing the subformulae.
The translation for fixed point operators is done by essentially starting a recursion on the subformula, meaning $\mu X \varphi$ is expanded to the state representing $\varphi$.
My question is now: why is it not allowed that such a least fix point operator is expanded infinitely often, while it would be acceptable for a greatest fix point operator?
