{\bf Abstract}

Isometries on formal power series over the finite field \ff_2
or on 2--adic integers can be
computed by invertible transducers on inputs from \{0,1\}^\infty.
We consider the structural complexity of an isometry f,
measured as {\it tree complexity} T(f,h), h the tree height
[H.~Niederreiter, M.~Vielhaber, {\it J.~Cpx.}, 12 (1996)]
and the computational complexity, as {\it number of bit operations}
B(f,n) needed for the first n input / output symbols.

We introduce the shift commutator
\SC(f) := \sigma^{-1}\circ f^{-1}\circ \sigma\circ f (\sigma the shift on
\{0,1\}^\infty)
and show\ \
that f\mapsto \SC(f) is a selfmap on the set of all isometries.
We obtain the polynomial bounds T(\SC(f),h) \leq T(f,h)^2 - T(f,h) + 2
and B(f,n) \leq n\cdot B(\SC(f),n)), by simulating f by n copies of
\SC.

On the other hand, trying to bound T(f,h) by T(\SC(f),h)
it turns out that {\it e.g.} for the isometries
connected to the continued fraction expansion and
to Collatz' 3N+1 conjecture, the function f itself is structurally
{\it exponentially} more complex than its \SC(f). Hence simulating f by
\SC(f) may yield sharper upper bounds for the bit complexity as can be
inferred from f alone.

We finish with some dynamical aspects of isometries like orbits, ergodicity,
preservation of measure.