Pesin

Pesin is a surname associated with several individuals, most notably Yakov B. Pesin, a Russian mathematician whose work laid the foundations of Pesin theory in dynamical systems. Pesin theory studies systems with nonuniform hyperbolicity, extending the classical theory of uniformly hyperbolic systems. Central ideas include Lyapunov exponents, which quantify average rates of separation of nearby trajectories, and the construction of local stable and unstable manifolds at almost every point with nonzero exponents (the Pesin stable/unstable manifold theorem). The theory also introduces Pesin blocks—sets on which hyperbolic behavior is uniform on fixed scales—facilitating a detailed analysis of invariant measures.

A key result is Pesin's entropy formula, which connects measure-theoretic entropy h_mu(f) to the sum of positive Lyapunov exponents, typically expressed as h_mu(f) = ∫ sum_{χ_i>0} χ_i(x) dmu(x). Together, these ideas provide a framework for understanding chaotic dynamics in smooth systems beyond the strictly uniform case, and have influenced ergodic theory, smooth dynamical systems, and applications in mathematical physics.

Pesin's work remains foundational in the study of nonuniform hyperbolicity and entropy, and his theories continue to guide research on the geometric structure of chaotic attractors and the statistical properties of dynamical systems.