Sinai–Ruelle–Bowen measure

Let ${\displaystyle T:X\rightarrow X}$  be a map. Then a measure ${\displaystyle \mu }$  defined on ${\displaystyle X}$  is an SRB 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]

${\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{i=0}^{n}\varphi (T^{i}x)=\int _{U}\varphi \,d\mu }$ 

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

One can see the SRB measure ${\displaystyle \mu }$  as one that satisfies the conclusions of Birkhoff'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 ${\displaystyle T:X\rightarrow X}$  be a ${\displaystyle C^{2}}$  diffeomorphism with an Axiom A attractor ${\displaystyle {\mathcal {A}}\subset X}$ . Assume that this attractor is irreducible, that is, it is not the union of two other sets that are also invariant under ${\displaystyle T}$ . Then there is a unique Borelian 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 the unstable manifold and submanifolds thereof;
3. ${\displaystyle h(T)=\int \log {{\bigl |}\det(DT)|_{E^{u}}{\bigr |}}\,d\mu }$ , where ${\displaystyle h}$  is the Kolmogorov–Sinai entropy, ${\displaystyle E^{u}}$  is the unstable manifold and ${\displaystyle D}$  is the differential operator.

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

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

${\displaystyle \lim _{\varepsilon \rightarrow 0}P_{\varepsilon }(\cdot \mid x)=\delta _{Tx}(\cdot ),}$ 

where ${\displaystyle \delta }$  is the Dirac 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