We investigate the relationship between endomorphisms of the Cuntz algebra
O2 and endomorphisms of the Thompson groups F, T and V
represented inside the unitary group of O2. For an endomorphism
λu of O2, we show that λu(V)⊆V if and
only if u∈V. If λu is an automorphism of O2 then
u∈V is equivalent to λu(F)⊆V. Our investigations are
facilitated by introduction of the concept of modestly scaling endomorphism of
On, whose properties and examples are investigated.Comment: v1: 10 pages. v2: minor changes, updated reference list, 11 pages; to
appear in Studia Mathematic