Non-uniform hyperbolicity

Non-uniform hyperbolicity is a generalization of hyperbolicity in which a dynamical system has positive and negative Lyapunov exponents at almost every point, but the rates of expansion and contraction vary across phase space rather than being bounded by uniform constants. This is the typical behavior in real-world systems — from the Hénon map to billiards to geodesic flows — and it requires the more delicate machinery of Pesin theory to establish the existence of stable and unstable manifolds. Unlike uniform hyperbolicity, non-uniform hyperbolicity does not guarantee structural stability or finite Markov partitions, but it does preserve statistical regularity through the existence of SRB measures and the machinery of Young towers. It is the frontier of chaos theory: the regime where rigidity dissolves but order persists.