Linear response theory says: push gently, get a proportional response. Apply a small bias to a chaotic system, and the average drift should be proportional to the force. This works beautifully for ergodic systems, for gases, for most of statistical mechanics.
Vijayan et al. show it fails when the chaos has hierarchical structure. In uniformly hyperbolic systems with progressively finer transport channels — nested scales of trapping and escape — reducing the bias doesn't reduce the response proportionally. Instead, smaller forces activate additional fine-scale transport channels that were invisible at larger forces. The effective mobility diverges as the bias approaches zero.
The mechanism: hierarchy creates scale-dependent permeability. At strong bias, only the coarse channels matter — the particle barrels through. At weak bias, the particle explores finer and finer channels, each contributing to transport. The number of contributing channels increases without bound as the force decreases. The gentler you push, the more of the system you engage.
This is not intermittency, not noise, not the usual suspects. The system is uniformly hyperbolic — as strongly chaotic as a dynamical system can be. Linear response breaks down not because the chaos is wild but because the architecture is nested. Hierarchy itself is the independent cause.
The structural claim: the relationship between perturbation and response depends on the geometry of the phase space, not just the dynamics on it. Two systems with identical Lyapunov exponents, identical mixing rates, identical statistical properties can have completely different transport responses if one is hierarchical and the other is not. The response encodes the architecture, not the dynamics.