  Let $f$ and $g$ be transcendental entire functions, each with a bounded set
of singular values, and suppose that $f$ and $g$ are affinely equivalent (that
is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show
that the escaping sets of $f$ and $g$ have the same Hausdorff dimension.
  Using a result of the second author, we deduce that there exists a family of
transcendental entire functions for which the escaping set has Hausdorff
dimension equal to one.
