We investigate the relationship between endomorphisms of the Cuntz algebra ${ \mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${ \mathcal O}_2$. For an endomorphism $\lambda_u$ of ${ \mathcal O}_2$, we show that $\lambda_u(V)\subseteq V$ if and only if $u\in V$. If $\lambda_u$ is an automorphism of ${ \mathcal O}_2$ then $u\in V$ is equivalent to $\lambda_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${ \mathcal O}_n$, whose properties and examples are investigated.
Categories: math.OA math.GR