# Homeomorphism with many fixed points

I am looking for some example (other than identity) of homeomorphism in the Torus $\mathbb{T}^2$ so that it has a set of fixed points with non-empty interior. It's possible?

I appreciate any reference.

• Intuitively there is a continuum of such homeomorphisms. May 30, 2017 at 4:22

Pick the compact set $F \subset \mathbb{T}$ of points that you want fixed (making sure it has open interior, or any other property you require). Perturb the identity so that any point in the complement of $F$ moves by a vector of magnitude proportional to its distance to $F$.

Depending on your choice for $F$, and vector directions, it should not be difficult to ensure your perturbed map is a homeomorphism. For instance:

• $F$ is an annulus, so the complement is an annulus too, and the vector can point in the direction of the core loop.
• $F$ is the union of a meridian and a parallel of $\mathbb{T}$. Then the complement is a topological disk, and the vector can point in the direction of a specific boundary point.
• The key point being how to «Perturb the identity so that…», no? May 30, 2017 at 5:10
• I gave two examples... May 30, 2017 at 5:13
• A systematic way of doing this is to pick a continuous function which vanishes on the set $F$, multiply it by a nowhere vanishing vector field and consider the function I constructed. May 30, 2017 at 5:25

Think first about the map $f:\mathbb{C}\rightarrow\mathbb{C}$ given by $$f(re^{i\theta})=(\min\{r, r^2\})e^{i\theta}$$ (for $\theta$ real and $r$ positive and real). This map is the identity outside the unit disc, and "stretches" the inside of the unit disc; and is an autohomeomorphism of $\mathbb{C}$ with the usual topology.

We can apply this same picture to the torus. Specifically, let $S$ be the set of complex numbers whose real and imaginary part each lie in $[-2, 2]$; this is a square of side length $4$ centered at the origin. The torus can be thought of as the quotient gotten by identifying opposite edges in $S$. Then the map $f$ described above transfers to an autohomeomorphism of the torus which is the identity on a set with nonempty interior, and is non-identity on a set with nonempty interior.

By changing the exponent, we get lots of similar examples; and of course this isn't the only way to do this sort of thing, by a long shot.

Pick any vector field $X$ on the torus $T$ which vanishes on an open set, and consider the "time 1" map for the flow it generates, that is, find the function $\Phi:\mathbb R\times T\to T$ such that $\Phi(0,x)=x$ for all $x\in T$ and $\frac{\mathrm d}{\mathrm dt}\Phi(t,x)=X_{\Phi(t,x)}$ and consider the function $f:x\in T\mapsto \Phi(1,x)\in T$, which is a diffeomorphism. Its fixed point set contains the vanishing locus of the field $X$.

The torus contains a homeomorphic copy of a closed disc together with an injection map $i: D\rightarrow\mathbb{T}^2$. Now take any non-trivial homeomorphism of a closed disc $\varphi: D\rightarrow D$ that fixes $\partial D$. For example $z\mapsto e^{i(1-|z|)\theta}z$ for some $\theta\neq 2k\pi, k\in\mathbb{Z}$. We can now construct our homeomorphism of the torus taking $i\circ\varphi\circ i^{-1}$ on $i(D)$ and identity on $\mathbb{T}^2\setminus i(D)$.