Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with that map in the sense, that $s(f(x),f(y),t)=f(s(x,y,t))$. Then one can ask, whether any self-homeomorphism of $X$, which is compatible with $s$ in the upper sense is already a isometry.

This is clearly wrong for $X=\mathbb{R}^n$, as all affine maps are compatible with $s$. So the question is, whether these are the only examples.

For example I think I can show, that the $n$-dimensional hyperbolic space ($n\ge 2$) is rigid in that sense.

Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with that map in the sense, that $s(f(x),f(y),t)=f(s(x,y,t))$. Then one can ask, whether any self-homeomorphism of $X$, which is compatible with $s$ in the upper sense is already a isometry.

This is clearly wrong for $X=\mathbb{R}^n$, as all affine maps are compatible with $s$. So the question is, whether these are the only examples.

For example I think I can show, that the $n$-dimensional hyperbolic space ($n\ge 2$) is rigid in that sense.

EDIT: Due to the big amout of counterexamples one could better ask the following question:

Are the spaces $\mathbb{R}^n$ the only spaces, which have self homeomorphisms compatible with $s$ (in the upper sense), that are not self-similarities ?