Oseledets

Slav warrior from Solntsev book

Oseledets theorem, also known as the Multiplicative Ergodic Theorem, is a fundamental result in the theory of dynamical systems and ergodic theory. It was proven by the Russian mathematician Valery Oseledets in 1968. This theorem provides a detailed analysis of the behavior of Lyapunov exponents for a dynamical system, which are measures of the rates of separation of infinitesimally close trajectories of the system. Understanding these exponents is crucial for analyzing the stability and chaotic behavior of dynamical systems.

Statement of the Theorem

The Oseledets theorem applies to a measure-preserving dynamical system \((X, \mathcal{B}, \mu, T)\) where \(X\) is a space, \(\mathcal{B}\) is a sigma-algebra, \(\mu\) is an invariant measure, and \(T: X \rightarrow X\) is a measurable transformation. Given a matrix-valued function \(A: X \rightarrow GL(d, \mathbb{R})\) that is integrable, the theorem asserts that there exists a full measure set \(X' \subset X\) such that for every \(x \in X'\), there are numbers \(\lambda_1 > \lambda_2 > \cdots > \lambda_k\) (the Lyapunov exponents), and a decomposition of \(\mathbb{R}^d\) into subspaces \(V_1(x), V_2(x), \ldots, V_k(x)\) that are invariant under \(A(x)\), with the property that for any nonzero vector \(v_i\) in \(V_i(x)\), the limit \[ \lim_{n \to \infty} \frac{1}{n} \log \|A(T^{n-1}x) \cdots A(x)v_i\| = \lambda_i \] exists and is equal to the corresponding Lyapunov exponent.

Implications and Applications

The Oseledets theorem has profound implications in various fields such as statistical physics, control theory, and mathematical finance. It provides a rigorous framework for understanding the stability of systems under small perturbations, the predictability of chaotic systems, and the growth rates of products of random matrices.

In statistical physics, the theorem helps in the study of disordered systems and the nature of phase transitions. In control theory, it aids in the design of stable control systems and the analysis of system robustness. In mathanical finance, it is used to model the growth rates of investments in random environments.

See Also

• Lyapunov exponents
• Dynamical systems
• Ergodic theory
• Measure-preserving transformation