Beyond Lyapunov Exponents: Transport, Ergodicity, and the Geometry of Chaotic Failure

Beyond Lyapunov Exponents: Transport, Ergodicity, and the Geometry of Chaotic Failure (Doug Doucette, June 2026) is a foundational monograph that reframes chaos from a purely local instability phenomenon into a transport-geometric one.

The central thesis is that a positive Lyapunov exponent only certifies point-prediction failure (nearby trajectories separate exponentially). It says nothing about whether the system can cross barriers, diffuse across action space, or escape a coherent operating region. Physical or operational failure occurs only when local chaos becomes transport-effective.

The work introduces the thin vs. thick chaos distinction, quantified by the thinness ratio

• Thin chaos ($\mathcal{T}\ll 1$): Genuine local instability exists, but transport remains confined by invariant tori, cantori, spectral gaps, or bounded chaotic components. Point prediction fails, yet envelope prediction (confinement) survives. Many real systems (Trojans, mode-locked lasers, localized molecular vibrations) live in this regime for long times.
• Thick chaos ($\mathcal{T}\gtrsim 1$): Local instability connects into a transport network (via resonance overlap, barrier leakage, spectral-gap collapse, or graph percolation). The chaotic component reaches the failure scale; envelope prediction collapses.

Rigorous propositions establish that Lyapunov positivity does not imply transport, that bounded chaotic components are automatically transport-thin, and that thick chaos is equivalent to the existence of a dynamical path from the coherent region to the failure region.

The strongest conclusion is concise and operational:

Failure begins not when chaos appears, but when chaos becomes connected.

This work therefore replaces the blunt question “Is the system chaotic?” with the sharper diagnostic pair: “Is there local instability?” and “Has that instability acquired transport connectivity relative to the failure scale?” This distinction supplies a geometry-based early-warning architecture that is strictly finer than classical Lyapunov or ergodic criteria alone, while remaining fully compatible with KAM theory, Nekhoroshev estimates, and Hamiltonian transport.