# Oseledets theorem

## Introduction

Let \(A_0,A_1, ... , A_t, ...\) be a sequence of nonsingular \( m\times m \)matrices satisfying \( \frac{1}{t} \log||A_{t}|| \rightarrow 0, \) and let \( A(t)=A_{t-1}\cdots A_0 \) for \( t=1,2,... \ .\) Suppose there is a number \(c > 0 \) such that \( ||A(t)||\le \exp(ct) \ .\) The Lyapunov exponent of a nonzero vector \(e\in R^m\) is defined by \[\tag{1} \chi(e)= \limsup\frac{1}{t}\log||A(t)e|| . \]

More generally, let \( e^k \) be a subspace of\(R^m\) of dimension \(k\) and \(\lambda(t,e^k)\) the absolute value of the determinant of the linear transformation \(e^k \rightarrow A(t)e^k\) defined by the matrix \(A(t)\ .\) In particular, \(\lambda(t,e^{1})=\frac{||A(t)e||}{||e||}\) for a nonzero vector \(e\in e^1\ .\) The Lyapunov exponent of the subspace is defined by \( \chi(e^{k})= \limsup\frac{1}{t}\log\lambda(t,e^k)\ .\) This reduces to the definition (1) for the one-dimensional case.

The function \(\chi(e)\) attains at most m distinct values \(\chi_1 < \chi_2 < .. < \chi_r \) for some \( r\le m \ .\) Let \( L^{i}\) be the subspace defined by the condition \( \chi(e)\le \chi_i, 0\ne e\in L^i .\) We have that \(L^0 ={0}\subset L^1 \subset ....\subset L^r = R^m \) and that \(\chi(e)=\chi_i \) for \(0\ne e\in L^i \setminus L^{i-1}, i=1,.. r.\)The number \(k_i= \dim(L^{i})-\dim(L^{i-1})\) is called the multiplicity of the value \(\chi_i\ .\) The sequence A(t) is said to be **Lyapunov regular** if
\[
\sum_{i=1}^r k_{i}\chi_i = \lim\frac{1}{t}\log| \det A(t)|.
\]

**Theorem 1.** If the sequence \(A(t)\) is Lyapunov regular, then the Lyapunov exponents of all orders are exact, i.e., \(
\chi(e^{k})=\lim\frac{1}{t}\log\lambda(t,e^k).
\)

Denote the transpose of the matrix \(A\) by \(A^*\ .\)

**Theorem 2.** If the sequence \(A(t)\) is Lyapunov regular, then the following is true : i) \(\lim(A^{*}(t)A(t))^{\frac{1}{2t}}=\Lambda \) where \(\Lambda \) is a diagonal matrix; ii) \(\exp(\chi_{1}),... ,\exp(\chi_{r})\) are the distinct eigenvalues of \(\Lambda\) and \(k_i\) the multiplicity of \(\exp(\chi_{i}) ;\) iii) \(\lim\frac{1}{t}\log||A(t)\Lambda^{-t}|| = 0.\)

## Oseledets' Multiplicative Ergodic Theorem

Let T be a measure preserving transformation of a probability Lebesgue space \((X,\Sigma,\mu)\) and \(A(t,x)=A(T^{t-1}x)...A(x)\ ,\) where \(A: X\rightarrow Gl(m,R)\) is a measurable map satisfying \(\log^{+}||A(x)|| \in L^{1}(X,\mu).\)

**Theorem 3.** The function \(t\rightarrow A(t,x) \) is Lyapunov regular for \(\mu\)-almost every \(x \ .\) The function \( \Lambda=\Lambda(x) \) is measurable. The filtration \( L^{1}(x) \subset ....\subset L^{r}(x) = R^m\)
is measurable.

## The case of invertible transformations

Let \(T\) and \(T^{-1}\) be measure preserving transformations. Assume that \(\log^{+}||A(x)||\) and \(\log^{+}||A^{-1}(x)||\) are integrable. Let \(A(t,x)=A^{-1}(T^{t}x)...A^{-1}(T^{-1}x),t\le -1 .\)

**Theorem 4.** The function \(t\rightarrow A(t,x) \) is Lyapunov regular as \(t\rightarrow \pm\infty \) for\(\mu\)-almost every \(x \ .\) There is a measurable splitting \(R^m = E^{k_{1}(x)}\oplus ...\oplus E^{k_{r}(x)}\)such that \( \lim_{t\rightarrow\pm\infty}\frac{1}{t}\log||A(t,x)e||=\chi_{i}(x) \) for \(0\ne e\in E^{k_{i}(x)} \) and \( \dim(E^{k_{i}(x)})=k_{i}(x). \) If \(e^k \subset E^{k_{i}(x)} \) then \(\lim_{t\rightarrow\pm\infty}\frac{1}{t}\log\lambda(e^{k})=k\chi_{i}(x)\) uniformly over \(e^k \subset E^{k_{i}}(x).\) Furthermore, \(\lim_{t\rightarrow\pm\infty}\frac{1}{t}\log\sin(\angle(E^{k_i}(T^{t}x),E^{k_j}(T^{t}x))=0, i\ne j \) and \(\chi_{i}(Tx)=\chi(x),k_{i}(Tx)= k_{i}(x),E^{k_{i}(Tx)}(Tx)=A(x)E^{k_{i}(x)}(x). \) The subspaces \(E^{k_{i}(x)} \) are called Oseledets subspaces.

## The continuous-time case

\(A(t,x)\) is called a cocycle if \(A(t+s,x)=A(t,T^{s}x)A(s,x),\) where \( A : R\times X\rightarrow GL(m,R)\) is a measurable function and \(\{T^{t}\}\) is a measurable measure preserving flow in a probability Lebesgue space \((X,\Sigma,\mu)\ :\) \(T^{t+s}=T^{t}T^{s}\ .\) Let \(\sup\{\log^{+}||A(t,x)|| : -1\le t\le 1\}\) be integrable. The statements such as in theorems 3,4 are also true in the continuous-time case. The derivatives of deterministic and stochastic flows provide examples of such cocycles.

## History

In 1965 the author of this paper was a graduate student. His scientific adviser was Y. Sinai. From Sinai's work it became clear that the positive entropy in the classical dynamical systems is related to exponential divergence of orbits originating at nearby points. This connection became the starting point of the author's interest in the problem of exponential divergence. In 1965, during the workshop on ergodic theory in Khumsan, author proved the multiplicative ergodic theorem (MET). The main idea of the proof of the MET is to reduce the general case to the case of triangular cocycles. In 1966 the author gave a talk entitled "The strong law of large numbers for random matrix processes" at the International Congress of Mathematicians in Moscow. A year later the author defended his Ph.D. thesis; the third chapter of this thesis was called "A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems". Finally, in 1968 the paper with the same title was published. Raghunathan gave another proof of the MET. It exploits Kingman's subadditive ergodic theorem. V.A. Kaimanovich proved a MET for semisimple Lie groups. Ruelle extended the MET to the case of Hilbert spaces and Mane extended it to the case of Banach spaces. Following V.A. Kaimanovich, A.Karlsson and G.Margulis obtained an extension of the MET to some nonpositively curved spaces. A.Karlsson and F.Ledrappier proved a MET for the group ISO(X) of a proper metric space X. Duchin proved a MET for the mapping class groups. You can find more details in the books by L. Arnold, by U. Krengel and by L. Barreira and Ya. Pesin for a description of various versions of the MET.

## Acknowledgments

The author's research on the MET was partially supported by RFBR Grant 07-01-00203.

## References

- A.M. Lyapunov, The general problem of the stability of motion, Taylor & Francis, 1992.
- V.I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231. Moscov.Mat.Obsch.19 (1968), 179-210.
- M.S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel JM 32 (1979), 365-362.
- D Ruelle, Ergodic theory od differentiable dynamical systems, Inst.des Hautes Etudes Scient., Publ. Math.50 (1979), 275-306.
- R. Mane, Lyapunov exponents and stable manifolds for compact transformations, Geometric dynamics,Lecture Notes in Math.1007, Springer (1983), 522-577.
- L. Arnold, Random dynamical systems, Monographs in Mathematics, Springer, 1998.
- U. Krengel, Ergodic theorems, Walter de Gruyter, Berlin New York, 1985.
- L. Barreira and Ya. Pesin, Nonuniform hyperbolicity: dynamics of systems with nonzero Lyapunov exponents, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007
- A. Karlsson and G.Margulis, A multiplicative ergodic theorem and nonpositive curved spaces, Comm.Math.Phys. 22 (1999), 107-123.
- V.A. Kaimanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups, J. Soviet Math. 47 (1989), 2387-2398.
- A. Karlsson and F. Leddrapier, On laws of large numbers for random walks, Annals of Probability 34 (2006), 1693-1706.
- M. Duchin, The thin triangles and a multiplicative ergodic theorem for Teichmuller geometry (2005), arXiv: math/0508046.

**Internal references**

- Edward Ott (2008) Attractor dimensions. Scholarpedia, 3(3):2110.

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

- Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.

- Yakov Pesin and Boris Hasselblatt (2008) Nonuniform hyperbolicity. Scholarpedia, 3(1):4842.

- Boris Hasselblatt and Yakov Pesin (2008) Pesin entropy formula. Scholarpedia, 3(3):3733.

## See Also

Dynamical Systems, Entropy, Ergodic Theory, Invariant Measures, Nonuniform Hyperbolicity, Partial Hyperbolicity, Pesin Entropy Formula