What is the limit of the sequence of iterated cosines?

Question

I asked this question on MSE here.

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Define $f_1(z) = \cos(z)$, $f_{n+1}= \cos(f_n (z)) $, The question is: Does $\lim\limits_{n \to \infty}f_n(z)$ exist for certain $z \in \mathbb{C}$? And what is the limit for such $z$?

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We know that for real numbers $z\in \mathbb{R}$, the limit exists and is the solution to the equation $x=\cos(x)$, this result is elementary. However, the complex case seems more intricate due to the unbounded nature of the cosine function on $\mathbb{C}$ and the existence of infinitely many solutions to $\cos(z)=z$.

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This is the graph of $f_{200}(z)$:

It seems that if the limit exists then the limit is the real solution for $\cos(z)=z$, For some reason it seems that the other fixed points of $\cos(z)$ don't "attract" the sequence of points, only the real solution do.

Answer 1

You should have a look at Bob Devaney's notes entitled Complex Exponential Dynamics, which you can download from his web page:

http://math.bu.edu/people/bob/papers.html

While that paper focuses on the exponential, there are some more general results for transcendental functions as well. In particular, Theorem 2.12 states that any attracting periodic cycle must attract some singular value. A singular value is either a critical value (i.e. image of a critical point) or an asymptotic value (i.e. the limiting value of $f(z(t))$, where $z(t)$ is a curve that tends to infinity).

Now, the cosine function has no asymptotic values just two critical values, namely $\pm1$. But the orbits of $\pm1$ converge after one iteration, since $\cos(1)=\cos(-1)$ due to the even symmetry of the cosine. As a result, there can be at most one attractive cycle under iteration of the complex cosine.

You might also find this question on Math.StackExchange interesting as well.

Answer 2

Cosine has one attracting fixed point $a\approx0.7390851$, 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.