The formula nu x. !p &<><>(x &p & phi_inf) & phi_inf does not make sense: If you forget about phi_inf at first, your formula becomes nu x. !p & <><>(x&p). I guess you rather mean nu x. p & <><>(x&p); then a fixpoint x must satisfy x = p & <><>(x&p). Thus, x must be a subset of p, but then x&p = x and therefore, you can write nu x. (p & <><>x). That should be the right formula. If you unroll it, you find
p & <><>(p & <><>(p & <><>x)) = p & <><>p & <><><><>p & <><><><><><>x
A path where whenever p holds, then p also holds next is not just a persistence property, since persistence properties allow a property for a while to do whatever, and then just from a sudden enforce that p must hold forever. Here, we would rather need EG(p->Xp) which is a CTL* formula, but not in CTL. µ-calculus makes a bit effort here, also.