SRB Measures and Attractors — Mathematics Knowledge System

Overview

A Sinai-Ruelle-Bowen (SRB) measure is an invariant probability measure that describes the long-run statistics of Lebesgue-typical orbits of a chaotic dynamical system. For a strange attractor, Lebesgue measure is not invariant (the attractor has measure zero), but most initial conditions in its basin of attraction converge to the same statistical behaviour described by the SRB measure. SRB measures are the physically relevant invariant measures: they are what observers see in experiments and numerical simulations.

Intuition

Imagine running a chaotic system (like the Lorenz system) from many different starting points in the basin of attraction. Despite the sensitivity to initial conditions, the time-averaged statistics -- the histogram of positions, the average energy -- converge to the SAME distribution for almost every starting point. This limiting distribution is the SRB measure. It is supported on the strange attractor (which has Lebesgue measure zero) but captures everything physical about the system.

Formal Definition

Definition

An invariant probability measure mu for a map f: M -> M is an SRB measure (or physical measure) if there exists a positive Lebesgue measure set B (basin of attraction) such that for Lebesgue-a.e. x in B: (1/n) sum_{k=0}^{n-1} delta_{f^k(x)} -> mu weak*. Equivalently, mu has absolutely continuous conditional measures on unstable manifolds (its disintegration along W^u directions is absolutely continuous).

\frac{1}{n}\sum_{k=0}^{n-1} \phi(f^k(x)) \xrightarrow{n\to\infty} \int \phi\, d\mu \quad \text{for Lebesgue-a.e. } x \in B

Ergodic averages converge to SRB measure

\mu|_{W^u} \ll \mathrm{Leb}_{W^u}

Absolute continuity on unstable manifolds

h_\mu(f) = \int \sum_{\lambda_i > 0} \lambda_i\, d\mu

Pesin entropy formula for SRB measures

Notation

Notation	Meaning
$\mu_{\mathrm{SRB}}$	SRB / physical invariant measure
$W^u(x)$	Unstable manifold at x
$h_\mu(f)$	Metric entropy of f with respect to mu
$\lambda_i$	Lyapunov exponents of the system

Theorems

Theorem 1: Sinai-Ruelle-Bowen Theorem

For an Axiom A attractor Lambda of a C^2 diffeomorphism f, there exists a unique SRB measure mu on Lambda. This measure is ergodic, has absolutely continuous conditional measures on unstable manifolds, and describes the statistics of Lebesgue-a.e. orbit in the basin of attraction: time averages converge to space averages under mu.

Theorem 2: Pesin Entropy Formula

For an SRB measure mu, the Kolmogorov-Sinai entropy equals the sum of positive Lyapunov exponents: h_mu(f) = integral (sum of positive Lyapunov exponents) dmu. This formula characterises SRB measures among all invariant measures: mu is SRB iff the Pesin formula holds as an equality.

Theorem 3: Uniqueness in Ergodic Decomposition

For a transitive Axiom A attractor, the SRB measure is unique and ergodic. For non-uniformly hyperbolic systems, SRB measures may not always exist or be unique, but when they exist, the Pesin formula and weak* basin convergence characterise them.

Worked Examples

1
The doubling map is uniformly expanding with Lyapunov exponent lambda = log 2 > 0 everywhere.
2
Lebesgue measure is invariant: f preserves Lebesgue measure because f^{-1}([a,b]) consists of two intervals each of half the length.
$\mathrm{Leb}(f^{-1}([a,b])) = \mathrm{Leb}([a,b])$
3
Lebesgue measure is ergodic for the doubling map (a classical result), and has absolutely continuous conditional measures on the (trivial) unstable manifolds.
4
By the Pesin formula: h_mu(f) = log 2 = integral lambda dmu = log 2. The formula holds, confirming Lebesgue measure is the SRB measure.

✓ Answer

Lebesgue measure is invariant and ergodic for the doubling map, satisfies the Pesin entropy formula, and is the SRB measure -- time averages converge to integrals against Lebesgue measure for a.e. starting point.

Practice Problems

Hardfree response

Distinguish between an SRB measure and an arbitrary invariant measure for a chaotic attractor. Why is the SRB measure the 'physically relevant' one?

MediumMultiple choice

Which property characterises SRB measures among all invariant measures?

They are supported on periodic orbitsThey satisfy the Pesin entropy formula: entropy = sum of positive Lyapunov exponentsThey have zero entropyThey are absolutely continuous with respect to Lebesgue measure on the whole manifold

Common Mistakes

Common Mistake

Thinking SRB measures are absolutely continuous with respect to Lebesgue measure on M.

SRB measures are typically supported on a strange attractor of zero Lebesgue measure. They are AC only on unstable manifold leaves, not on the full manifold.

Quiz

An SRB measure describes the statistics of:

Orbits starting on periodic pointsLebesgue-typical orbits in the basin of attractionOrbits with the largest Lyapunov exponentOrbits confined to the stable manifold

Historical Background

SRB measures were introduced independently by Sinai (1972) for Anosov diffeomorphisms, Ruelle (1976) for Axiom A attractors, and Bowen (1975) for uniformly hyperbolic systems. The theory was extended to non-uniformly hyperbolic systems (including the Lorenz attractor) by Benedicks-Young (1993) and Viana. SRB measures unified the ergodic theory of chaotic systems, connecting Lyapunov exponents, entropy, and physical observation.

1972
Sinai introduces SRB measures for Anosov diffeomorphisms
Yakov Sinai
1975
Bowen characterises Gibbs states for hyperbolic systems
Rufus Bowen
1976
Ruelle extends SRB theory to Axiom A attractors
David Ruelle
1993
Benedicks and Young prove existence of SRB measures for the Henon attractor
Michael Benedicks, Lai-Sang Young

Summary

SRB (physical) measures describe the long-run statistics of Lebesgue-typical orbits of a chaotic system.
They are invariant, ergodic, and have absolutely continuous conditional measures on unstable manifolds.
The Pesin entropy formula h_mu = sum of positive Lyapunov exponents characterises SRB measures.
Every Axiom A attractor has a unique SRB measure; existence for non-uniformly hyperbolic systems is harder to establish.

References

BookYoung, L.-S. What are SRB measures and which dynamical systems have them? Journal of Statistical Physics, 2002.
BookKatok, A. and Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems. Cambridge, 1995.
WebsiteWikipedia -- SRB measure