Statement of The Theorem
Let μ be an invariant measure on X and C a cocycle of the dynamical system such that for each t∈T, the maps and are L1-integrable with respect to μ. Then for μ-almost all x and each non-zero vector u∈Rn the limit
exists and assumes, depending on u but not on x, up to n different values. These are the Lyapunov exponents.
Further, if λ1 > ... > λm are the different limits then there are subspaces Rn = R1 ⊃ ... ⊃ Rm ⊃ Rm+1 = {0} such that the limit is λi for u∈ Ri\Ri+1 and i = 1, ..., m.
The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that g : X → X is a one-to-one map such that and its inverse exist then the values of the Lyapunov exponents do not change.
Read more about this topic: Oseledets Theorem
Famous quotes containing the words statement of the, statement of, statement and/or theorem:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“I think, therefore I am is the statement of an intellectual who underrates toothaches.”
—Milan Kundera (b. 1929)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)