Dynamical systems and ergodic theory Definition

Pesin theory

Builds on Smooth manifold Diffeomorphism Invariant measure Lyapunov exponent Oseledets theorem

Definition

Let $M$ be a compact Riemannian manifold and let $f \colon M \to M$ be a $C^{1+\alpha}$ diffeomorphism for some $\alpha > 0$ . Let $\mu$ be an $f$ -invariant Borel probability measure on $M$ . Let $x \in M$ and let $v \in T_x M$ be nonzero. The Lyapunov exponent of $v$ is $\lambda(x, v) = \lim_{n \to \infty} \frac{1}{n} \log \norm{D f^n_x \, v}$ when the limit exists. By the Oseledets theorem, for $\mu$ -almost every $x$ the limit exists for every $v$ and takes finitely many values. The measure $\mu$ is hyperbolic if, for $\mu$ -almost every $x$ , the number $0$ is not among the values $\lambda(x, v)$ . Pesin theory is the theory of the pairs $(f, \mu)$ of this kind with $\mu$ hyperbolic, also called nonuniformly hyperbolic systems.

Remarks

Example: let $f$ be the linear map of the torus $\R^2 / \Z^2$ given by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ , and let $\mu$ be Lebesgue measure. The exponents are $\log \frac{3 + \sqrt 5}{2} > 0$ and $\log \frac{3 - \sqrt 5}{2} < 0$ at every point, so $\mu$ is hyperbolic.

Non-example: an irrational rotation of the torus $\R^2/\Z^2$ with Lebesgue measure is an isometry, so both exponents are $0$ everywhere and the measure is not hyperbolic.

The main results of the theory, for hyperbolic $\mu$ and $C^{1+\alpha}$ maps, include the following. Local stable and unstable manifolds exist at $\mu$ -almost every point. If $\mu$ is absolutely continuous with respect to volume, the entropy of $\mu$ equals the integral of the sum of the positive exponents, counted with multiplicity (the Pesin entropy formula). The hypothesis $\alpha > 0$ is needed for the stable manifold theorem: the theory does not extend in general to $C^1$ maps.

Reading path

To understand this page from the ground up, read these 111 pages in order. The foundations come first.

Almost everywhere concept
Birkhoff sum and average concept
Conditional expectation concept
Construction of the reals from Cauchy sequences concept
Element definition
Ergodic transformation concept
Euclidean space object
Function concept
Hilbert space concept
Integers object
Integrable function concept
Invariant measure concept
Limit of a sequence concept
Lp space concept
Lyapunov exponent concept
Measure-preserving transformation concept
Operator norm concept
Ordered field definition
Orthogonal complement concept
Orthogonal projection concept
Probability space concept
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Smooth manifold concept
Unitary operator concept
Set definition
Total derivative concept
Function of several variables concept
Ring of integers statement
Maximal ergodic theorem statement
von Neumann mean ergodic theorem statement
Disjunction definition
Implication definition
Predicate definition
Real sequence definition
Infimum definition
Supremum definition
Ordered pair definition
Differentiable function of several variables concept
Union definition
Existential quantifier definition
Universal quantifier definition
Fekete's lemma statement
Sample space definition
The rationals form a field statement
Completeness axiom statement
Cartesian product definition
Diffeomorphism concept
Uniqueness quantifier definition
Subset definition
Existence of square roots of nonnegative reals statement
The rationals form an ordered field statement
Binary relation definition
Universal set definition
Power set definition
Order on the reals definition
Square root definition
Function definition
Complement definition
Interval definition
Injective function definition
Sigma-algebra definition
Preimage of a set definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Almost surely definition
Conditional probability definition
Random variable definition
Monotonicity of probability statement
Almost sure convergence definition
Discrete random variable definition
Independence of two events definition
Law of a random variable definition
Convergence in probability definition
Random variable with a density definition
Mutual independence of events definition
Joint probability distribution definition
Independence of two random variables definition
Expected value definition
Identically distributed random variables definition
Independent sigma-algebras definition
Marginal distribution definition
Variance definition
Linearity of expectation statement
Markov's inequality statement
Independence of random variables definition
Independence means the joint law is the product of the marginals statement
Covariance definition
Standard deviation definition
Expectation of a product of independent random variables statement
Independent and identically distributed random variables definition
Variance of a sum statement
Chebyshev's inequality statement
Law of large numbers concept
Independent random variables are uncorrelated statement
Variance of a sum of independent random variables statement
Weak law of large numbers statement
Strong law of large numbers statement
Birkhoff's ergodic theorem statement
Kingman's subadditive ergodic theorem statement
Oseledets theorem statement
Pesin theory definition

Tag	`VM7M0PJTD4`
URI	`m2://VM7M0PJTD4`
Kind	definition
Writing	W0 stub
MSC	37D25, 37C40
Defines	Pesin theory
Uses	Smooth manifold, Diffeomorphism, Invariant measure, Lyapunov exponent, Oseledets theorem

Pesin theory

Definition

Remarks

Reading path

Further reading

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