Non-Hyperbolic Thermodynamics

Non-hyperbolic thermodynamics is the study of statistical descriptions for dynamical systems that lack the uniform expansion and contraction properties required by classical thermodynamic formalism. In non-hyperbolic systems — including the Hénon map, the Lorenz system, and many dissipative partial differential equations — invariant measures do not decompose cleanly into stable and unstable directions, and the transfer operator lacks the spectral gap that guarantees exponential decay of correlations.

The challenge is to define pressure, equilibrium measures, and entropy for systems where the symbolic dynamics requires an infinite grammar and the Newhouse phenomenon produces infinitely many coexisting attractors. Recent approaches use Young towers, inducing schemes, and Markov towers to construct thermodynamic descriptions for specific non-hyperbolic classes, but a general theory remains elusive.

Non-hyperbolic thermodynamics is the frontier where the beautiful island of classical thermodynamic formalism meets the sea of real-world complexity. Whether the island can be expanded or whether the sea is fundamentally different is the open question.