Cosine has one attracting fixed point $a\approx0.999847741$$a\approx0.7390851$, and both critical values $\pm1$ are attracted to it. Then it follows from general theorems of dynamics of entire functions that it has one completely invariant domain $D$ such that for all $z\in D$ trajectories converge to this attracting point. This domain is seen in your picture.
So for $z\in D$ the limit is $a$, and for all other $z$ the limit can exist only in the case when the trajectory stabilizes, that is if $z$ is a preimage of a repellng fixed point. Repelling fixed points are solutions of $\cos z=z$ different from $a$, and there are infinitely many of them. And each of them has infinitely many preimages which accumulate to $\partial D$, which is the Julia set. The region $D$ is dense in the plane. And preimages of the fixed points are dense on $\partial D$.
A reference for general theorems on dynamics of entire functions is
MR1216719 Walter Bergweiler, Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188.