Continuous surjective functions from the unit disk to itself that agree nowhere Do there exist two continuous surjective functions $f,g:D \to D$ such that $f(z) \neq g(z) $ for all $z \in D$, where $D$ is the closed unit disk? 
 A: Note: $D$ is now known to be the closed unit disk... I'll edit when I can.
I'll assume you mean $D$ to be the open unit disk.
Take 
$$
f(z)=z
$$
and 
$$
g(z)=\frac{z+1/2}{1+z/2}=\frac{2z+1}{z+2}.
$$
The latter is a biholomorphism of $D$ onto itself.
And it is easy to check that it has no fixed point in $D$.
A: Here is a example. Let $D$ be the closed unit disk of $\mathbb{C}$ and $D_L:=\{ z \in D | Re(z) \geq 0 \}$, $D_R:=\{ z \in D | Re(z) \leq 0 \}$ be the right half and the left half unit disk.
Let $f_1 : D \rightarrow D_L$ be an homeomorphism such that $f_1([-1,1])=[0,1]$ and such that $f_1$ sends the upper (resp. lower) half disk of $D$ homeomorphically to the upper (resp. lower) quater disk of $D_L$. Define an homeomorphism $g : D \rightarrow D_R$ in the same way.
Define
$$f_2 : D_L \rightarrow  D, z=r e^{i\theta} \mapsto z^2=r^2 e^{i2\theta},$$
$$g_2 : D_R \rightarrow  D, z=r e^{i(\pi-\theta)} \mapsto r^2 e^{i(\pi+2\theta)}.$$
You should think $f_1$ and $g_1$ as Horseshoe maps. Note $g_2$ is a horseshoe map composed with a symmetry (Make a picture !).
Now define $f,g : D \rightarrow D$ by $f = f_2 \circ f_1$ and $g = g_2 \circ g_1$.
It is obvious that $f$ and $g$ are continuous and onto. The functions $f$ and $g$ disagree everywhere because :


*

*$f$ sends the upper (resp. lower) half disk onto the upper (resp. lower) half disk.

*$g$ sends the upper (resp. lower) half disk onto the lower (resp. upper) half disk.

*$f$ sends $[-1,1]$ to $[0,1]$ with $f(-1)=0$.

*$g$ sends $[-1,1]$ to $[-1,0]$ with $g(1)=0$.

A: Hint: Look up the classification of elements of $SL(2;\mathbb{R})$. The answer will depend (but only slightly) on whether you are considering the open or closed unit disk.
Further hint: An element of $SL(2;\mathbb{R})$ gives an isometry of hyperbolic space. (Isometries are in particular continuous and surjective.) Hyperbolic space can be identified with the unit disk. Take one of your maps to be the identity. For the other map, it will suffice to find an isometry of hyperbolic space which fixes no points. This is where the classification of elements of $SL(2;\mathbb{R})$ is useful.
