Eckmann-Ruelle conjecture

The Eckmann-Ruelle conjecture is a central open problem in smooth ergodic theory, formulated by Jean-Pierre Eckmann and David Ruelle in 1985. It asserts that for a typical smooth dynamical system preserving a smooth measure, all Lyapunov exponents are non-zero almost everywhere, and the measure-theoretic entropy equals the sum of the positive Lyapunov exponents. This equality — a deterministic analogue of the Shannon-McMillan-Breiman theorem — would establish that chaotic systems produce information at exactly the rate predicted by their geometric instability.

The conjecture is known to hold for uniformly hyperbolic systems (where it follows from the work of Rufus Bowen and Yakov Sinai) and for certain non-uniformly hyperbolic systems (through the work of Jacob Pesin and others). But the general case — for typical diffeomorphisms of arbitrary manifolds — remains unproved. The difficulty lies not in computing Lyapunov exponents but in proving that they are non-zero: that typical systems are chaotic in almost every direction, almost everywhere.

If true, the conjecture would unify the geometric, probabilistic, and information-theoretic faces of chaos. It would imply that the only obstruction to statistical regularity in smooth dynamics is the absence of expansion and contraction — a condition so restrictive that it would exclude almost every interesting system.

The Eckmann-Ruelle conjecture is the audacious claim that chaos is not the exception but the rule, and that the information produced by a chaotic system is not a mystery but a theorem waiting to be proved. To settle it is to settle the question of whether the universe, at the scale of smooth maps, is fundamentally information-generating.

A closely related open problem concerns the Lyapunov spectrum of infinite-dimensional systems and whether the conjecture extends to coupled map lattices and other spatially extended models.