I am looking for a non-trivial example of a map $f:\mathbb{S}^n \to \mathbb{S}^n$ which is $L$-Lipschitz (w.r.t the intrinsic metric on the sphere or the Euclidean one, it does not matter), where $L < 1$.
By non-trivial I mean that $f$ is non-constant. If more "interesting" examples can be constructed, like maps which are not constant on any open subset this would be great.
Note that it's a theorem that the image of any such map $f$, must be contained in a hemisphere.
I will also be happy with examples which are only defined on $\mathbb{S}^n\setminus\{ N\}$;
If we remove a non-trivial "interval" from $\mathbb{S}^1$ for instance, it is not hard to construct contracting maps, by "pushing\squeezeing" the remainder towards the "opposite" hemicircle. However, this cannot be done for the domain $\mathbb{S}^1\setminus\{ N\}$ since we will have points which are arbitrarily closed, and it seems no finite "contracting factor" will suffice.
First reflect a hemisphere onto the other. This has Lipschitz constant $1$. Now it suffices to find an $L$-Lipschitz function from the hemisphere $H$ inside itself, $L<1$.
To achieve this you can do the following: say the hemisphere $H$ is "centered" at the pole $N$. Then:
Project orthogonally $H$ on the tangent space to $H$ at $N$. ($1$-Lipschitz)
Apply a homotety of factor $L<1$ to the tangent space. ($L$-Lipschitz)
Project on the closed ball $\overline B(0,1)$ ($1$-Lipschitz being the projection on a convex set)
After the last step the image of $H$ will be actually contained in $\mathbb S^{n}$, and the composition will be $L$-Lipschitz.