  For a jointly measurable probability-preserving action
$\tau:\mathbb{R}^D\curvearrowright (X,\mu)$ and a tuple of polynomial maps
$p_i:\mathbb{R}\to \mathbb{R}^D$, $i=1,2,...,k$, the multiple ergodic averages
\[ \frac{1}{T}\int_0^T (f_1\circ \tau^{p_1(t)})(f_2\circ \tau^{p_2(t)})...
(f_k\circ \tau^{p_k(t)})\,\mathrm{d} t \] converge in $L^2(\mu)$ as $T \to
\infty$ for any $f_1,f_2,...,f_k \in L^\infty(\mu)$. This confirms the
continuous-time analog of the conjectured norm convergence of discrete
polynomial multiple ergodic averages, which in is its original formulation
remains open in most cases. A proof of convergence can be given based on the
idea of passing up to a sated extension of $(X,\mu,\tau)$ in order to find
simple characteristic factors, similarly to the recent development of this idea
for the study of related discrete-time averages, together with a new inductive
scheme on tuples of polynomials. The new induction scheme becomes available
upon changing the time variable in the above integral by some fractional power,
and provides an alternative to Bergelson's PET induction, which has been the
mainstay of positive results in this area in the past.
