Dimension Bounds for Contractive Reservoir Computing from Input Entropy

Echo State Networks (ESNs) are designed to be \emph{stable}—the echo state property makes the reservoir state a well-defined function of the input history—but it remains unclear how \emph{geometrically rich} the reachable reservoir states are and what controls that richness. We give a partition-free, fractal-geometric theory of contractive reservoirs by viewing a quantized-input ESN as a contractive random dynamical system and, equivalently, an iterated function system (IFS). This identifies the reachable set as the IFS attractor and the stationary reservoir-state law under i.i.d.\ inputs as the unique invariant IFS measure. We prove entropy–contraction (entropy–Lyapunov) bounds on the intrinsic dimension of this measure, and in a conformal/similarity regime with standard separation conditions we obtain sharp closed-form dimension laws of the form “input entropy divided by contraction.” These results yield a quantitative \emph{criticality principle}: weakening contraction drives a transition from low-dimensional to full-dimensional state representations at an explicit threshold set by input entropy and contraction rates, providing practical design guidance for tuning leak/spectral radius and input scaling.