Let T : X → X {\displaystyle T:X\rightarrow X} be a map. Then a measure μ {\displaystyle \mu } defined on X {\displaystyle X} is an SRB measure if there exist U ⊂ X {\displaystyle U\subset X} of positive Lebesgue measure, and V ⊂ U {\displaystyle V\subset U} with same Lebesgue measure, such that:67
for every x ∈ V {\displaystyle x\in V} and every continuous function φ : U → R {\displaystyle \varphi :U\rightarrow \mathbb {R} } .
One can see the SRB measure μ {\displaystyle \mu } as one that satisfies the conclusions of Birkhoff's ergodic theorem on a smaller set contained in X {\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.8
Theorem 1:9 Let T : X → X {\displaystyle T:X\rightarrow X} be a C 2 {\displaystyle C^{2}} diffeomorphism with an Axiom A attractor A ⊂ X {\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 T {\displaystyle T} . Then there is a unique Borelian measure μ {\displaystyle \mu } , with μ ( X ) = 1 {\displaystyle \mu (X)=1} ,10 characterized by the following equivalent statements:
Also, in these conditions ( T , X , B ( X ) , μ ) {\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 μ {\displaystyle \mu } equals the zero-noise limit stationary distribution of a Markov chain with states T i ( x ) {\displaystyle T^{i}(x)} .11 That is, consider that to each point x ∈ X {\displaystyle x\in X} is associated a transition probability P ε ( ⋅ ∣ x ) {\displaystyle P_{\varepsilon }(\cdot \mid x)} with noise level ε {\displaystyle \varepsilon } that measures the amount of uncertainty of the next state, in a way such that:
where δ {\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,12 though nothing can be said about the amount of noise that is tolerable.
Walters, Peter (2000). An Introduction to Ergodic Theory. Springer. ↩
Bonatti, C.; Viana, M. (2000). "SRB measures for partially hyperbolic systems whose central direction is mostly contracting". Israel Journal of Mathematics. 115 (1): 157–193. doi:10.1007/BF02810585. S2CID 10139213. https://doi.org/10.1007%2FBF02810585 ↩
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Young, L. S. (2002). "What are SRB measures, and which dynamical systems have them?". Journal of Statistical Physics. 108 (5–6): 733–754. doi:10.1023/A:1019762724717. S2CID 14403405. /wiki/Doi_(identifier) ↩
If it does not integrate to one, there will be infinite such measures, each being equal to the other except for a multiplicative constant. ↩
Cowieson, W.; Young, L. S. (2005). "SRB measures as zero-noise limits". Ergodic Theory and Dynamical Systems. 25 (4): 1115–1138. doi:10.1017/S0143385704000604. S2CID 15640353. /wiki/Doi_(identifier) ↩