Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(x)=g(x)$?
If one of the mappings is invertible, then it is just a restatement of Brouwer's fixed point theorem but I do not know the answer in the general case and would not even dare to guess what it must be. Also, the answer is well-known to be "Yes" in dimension $1$.
A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf
The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.
Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.
Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf
I'm sure what I write has been thought of by many, but it's a starting point that I thought should be written down.
First by the Brouwer fixed point theorem $f$ has at least one fixed point, say $\bar{x}=f(\bar{x})$.
If that fixed point is unique (contraction mappings spring to mind for a bunch of examples of this) we're done since $g(\bar{x})=g(f(\bar{x}))=f(g(\bar{x}))$ and we see that $g(\bar{x})$ is "another" fixed point of $f$, since the fixed point was unique $g(\bar{x})=\bar{x}=f(\bar{x})$.
For "less nice" $f$ we still have that $f(g(\bar{x}))=g(\bar{x})$... in fact for any $n\in\mathbb{N}$ we have $f(g^n(\bar{x}))=g^n(\bar{x})$. If (without resorting to sequences) $g^n(\bar{x})\to y$, we can again claim success since we'll have $g(y)=y$ and $f(y)=y$.
Unless there's another "obvious" easy case I missed it seems like the interesting cases will be when $g^n(\bar{x})$ does not converge. Two sub-cases spring to mind: when $g^{n}(\bar{x})$ has finitely many accumulation points (like when $g^n(\bar{x})$ is a periodic point of $g$), or ... it has lots. Intuition (really thinking about rational and then irrational rotations about the origin as one way to generate those two cases) tells me that in either of these cases what we really need to do drop the $\bar{x}$ as a "starting point".
It "would be nice" if we can show $g$ conjugate to a rotation in the above two cases. Any thought on if that is true or not? I suspect not else $g$ would have a unique fixed point and we'd be done (as above)... Maybe semi-conjugate... but would that help? New minds, any thoughts?
I just ran across some information on especially the one-dimensional version of this problem, which appears in Victor Klee's unpublished Unsolved Problems in Intuitive Geometry. His formulation is:
"Suppose $I$ is a closed interval of real numbers, and $f$ and $g$ are commuting continuous maps of $I$ into itself. Must they have a common fixed point?"
He attributes the problem to J.R. Isbell from 1957 ["Research Problem #7: Commuting mappings of trees," Bull. Amer. Math. Soc. 63 (1957), 419]. Negative answers by Boyce and Huneke were published (in Notices Amer. Math. Soc.) in the late 1960's. (Harald Hanche-Olsen cites the Huneke paper in comments above.) Branko Grünbaum cites a recent survey on the topic: Eric L. McDowell, "Coincidence Values of Commuting Functions," Topology Proceedings 34 (2009) pp. 365-384. I cannot easily access this paper, but Grünbaum says it contains new results and an extensive bibliography.
There is also a result due to Shields which implies a common fixed point (and hence a coincidence point) assuming that $f$ and $g$ are both analytic on the disk.
A. L. Shields, On fixed points of commuting analytic functions, Proc. Amer. Math. Soc. 15(1964), 703-706. MR 29 #2790.
The following is not a solution but rather a reformulation of the original problem.
To make $f\colon B\to B$ invertible let us pass from the ball $B$ to the solenoid $S_f$, $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$
Map $f$ induces the shift $f_*\colon S_f\to S_f$ and g induces $g_*(\{x_n\})=\{g(x_n)\}$. They commute and $f_*$ is invertible. If Brower's fixed point theorem were true for $S_f$ the result would follow. Indeed, a fixed point of $f_*^{-1}\circ g_*$ gives the desired orbit.
Solenoid $S_f$ is a non-empty closed subset of $B^{\mathbb Z}$ eqiupped with the product topology. Hence $S_f$ is compact. In fact, $S_f\subset K^{\mathbb Z}$, where $K=\cap_{n>0}f^n(B)$. And I think it is plausible (can anybody give a proof?) that $S_f$ contracts to the orbit of the fixed point of $f$. (later: I am not that sure now)
Unfortunately, Brower's fixed point theorem doesn't hold for compact contractible spaces. But this seem to be quite old and active research area as I learned from the introduction to this paper http://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02071-1/S0002-9947-99-02071-1.pdf People prove positive results under various additional assumptions.
So, maybe it's useful to look at the problem from this point of view.
Off on a tangent ...
Kakutani's Fixed-Point Theorem A commuting family of continuous affine maps of a weakly compact convex set into itself has a common fixed point.
May be a possible way to solve the problem is to set up it in the context of closed relations instead of functions, by putting $R = g^{-1} \circ f : B^n \to B^n$, and assuming for instance $f(B^n) \subset g(B^n)$. So, we are looking for $x \in B^n$ such that $x \in R(x)$.
In shape theory there are defined the so-called multihomotopy groups, which generalize the classical homotopy groups by considering multi-valued loops instead of the single-valued ones. See http://plms.oxfordjournals.org/content/s3-69/2/330 and http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1226598673
Under nice circumstances the multihomotopy groups of spheres are $\tilde\pi_n(S^n) \cong \Bbb Z$, while for the $n$-disk are trivial.
One can try to adapt the proof of Brouwer's fixed point theorem in this setting.
So, assuming by contradicition that $f(x) \ne g(x)$ for all $x \in B^n$, one can consider the relation $T : B^n \to S^{n-1}$ where $T(x)$ is the intersection between $S^{n-1}$ and the half-lines starting from points of $R(x)$, and passing through $x$. It follows that $T$ is the identity on $S^{n-1}$, so $T$ is a multi-valued retraction $B^n \to S^{n-1}$, and this contradicts the fact that $\tilde\pi_{n-1}(S^{n-1})$ is not trivial.
I think that there are many details to fill, and that one should give the "nice" conditions under this approach works (for instance, assuming that $g^{-1}(y)$ is countable for all $y \in B^n$ does suffice?) but this could be an interesting application of multihomotopy groups.
I made a little google search and found this paper, that mentions that the conjecture holds true for polynomials and special functions Cohen calls "full functions". I'm not able to download the paper though. Cohen's paper is: H. Cohen, On fixed points of commuting functions, Proc. Amer. Math. Soc. 15. (1964),
An idea that occured to me, since apparently this is true for polynomials. Can we not use some form of approximation theorem (Stone-Weierstrass or whatever there is) to conclude the result for other continuous functions between the closed unit balls?
