  Given a target function $f$, how large must a neural network be in order to
approximate $f$? Recent works examine this basic question on neural network
\textit{expressivity} from the lens of dynamical systems and provide novel
``depth-vs-width'' tradeoffs for a large family of functions $f$. They suggest
that such tradeoffs are governed by the existence of \textit{periodic} points
or \emph{cycles} in $f$. Our work, by further deploying dynamical systems
concepts, illuminates a more subtle connection between periodicity and
expressivity: we prove that periodic points alone lead to suboptimal
depth-width tradeoffs and we improve upon them by demonstrating that certain
``chaotic itineraries'' give stronger exponential tradeoffs, even in regimes
where previous analyses only imply polynomial gaps. Contrary to prior works,
our bounds are nearly-optimal, tighten as the period increases, and handle
strong notions of inapproximability (e.g., constant $L_1$ error). More broadly,
we identify a phase transition to the \textit{chaotic regime} that exactly
coincides with an abrupt shift in other notions of function complexity,
including VC-dimension and topological entropy.
