In the mathematical discipline ofergodic theory, aSinai–Ruelle–Bowen (SRB) measure is aninvariant measure that behaves similarly to, but is not anergodic measure. In order to be ergodic, the time average would need to equal the space average for almost all initial states${\displaystyle x\in X}$, with${\displaystyle X}$ being thephase space.^[1] For an SRB measure${\displaystyle \mu }$, it suffices that the ergodicity condition be valid for initial states in a set${\displaystyle B(\mu )}$ of positiveLebesgue measure.^[2]

The initial ideas pertaining to SRB measures were introduced byYakov Sinai,David Ruelle andRufus Bowen in the less general area ofAnosov diffeomorphisms andaxiom A attractors.^[3][4][5]

Let${\displaystyle T:X\to X}$ be amap. Then a measure${\displaystyle \mu }$ defined on${\displaystyle X}$ is anSRB measure if there exist${\displaystyle U\subset X}$ of positive Lebesgue measure, and${\displaystyle V\subset U}$ with same Lebesgue measure, such that:^[2][6]

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{i=0}^n\phi (T^{i}x)={\int }_U\phi d\mu }$

for every${\displaystyle x\in V}$ and every continuous function${\displaystyle \phi :U\to \mathbb{R}}$.

One can see the SRB measure${\displaystyle \mu }$ as one that satisfies the conclusions ofBirkhoff's ergodic theorem on a smaller set contained in${\displaystyle X}$.

The following theorem establishes sufficient conditions for the existence of SRB measures. It considers the case of Axiom A attractors, which is simpler, but it has been extended times to more general scenarios.^[7]

Theorem 1:^[7] Let${\displaystyle T:X\to X}$ be a${\displaystyle C^2}$diffeomorphism with anAxiom A attractor${\displaystyle \mathcal{A}\subset X}$. Assume that this attractor isirreducible, that is, it is not the union of two other sets that are also invariant under${\displaystyle T}$. Then there is a uniqueBorelian measure${\displaystyle \mu }$, with${\displaystyle \mu (X)=1}$,^[a] characterized by the following equivalent statements:

1. ${\displaystyle \mu }$ is an SRB measure;
2. ${\displaystyle \mu }$ has absolutely continuous measures conditioned on theunstable manifold and submanifolds thereof;
3. ${\displaystyle h(T)=\int \log |det(DT){|}_{E^u}|d\mu }$, where${\displaystyle h}$ is theKolmogorov–Sinai entropy,${\displaystyle E^u}$ is the unstable manifold and${\displaystyle D}$ is thedifferential operator.

Also, in these conditions${\displaystyle \left(T,X,\mathcal{B}(X),\mu \right)}$ is ameasure-preserving dynamical system.

It has also been proved that the above are equivalent to stating that${\displaystyle \mu }$ equals the zero-noise limitstationary distribution of aMarkov chain with states${\displaystyle T^i(x)}$.^[8] That is, consider that to each point${\displaystyle x\in X}$ is associated a transition probability${\displaystyle P_{\epsilon }(\cdot \mid x)}$ with noise level${\displaystyle \epsilon }$ that measures the amount of uncertainty of the next state, in a way such that:

${\displaystyle \underset{\epsilon \to 0}{lim}{P}_{\epsilon }(\cdot \mid x)={\delta }_{Tx}(\cdot ),}$

where${\displaystyle \delta }$ is theDirac measure. The zero-noise limit is the stationary distribution of this Markov chain when the noise level approaches zero. The importance of this is that it states mathematically that the SRB measure is a "good" approximation to practical cases where small amounts of noise exist,^[8] though nothing can be said about the amount of noise that is tolerable.

• Quasi-invariant measure
• Krylov–Bogolyubov theorem
• Gibbs measure

• Ergodic theory

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