Sinai–Ruelle–Bowen measure

Let T:X \rightarrow X be a map. Then a measure \mu defined on X is an SRB measure if there exist U \subset X of positive Lebesgue measure, and V \subset U with same Lebesgue measure, such that: : \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i = 0}^n \varphi(T^i x) = \int_U \varphi \, d\mu for every x \in V and every continuous function \varphi: U \rightarrow \mathbb{R}. One can see the SRB measure \mu as one that satisfies the conclusions of Birkhoff's ergodic theorem on a smaller set contained in X. ==Existence of SRB measures==

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. Theorem 1: That is, consider that to each point x \in X is associated a transition probability P_\varepsilon(\cdot \mid x) with noise level \varepsilon that measures the amount of uncertainty of the next state, in a way such that: : \lim_{\varepsilon \rightarrow 0} P_{\varepsilon}(\cdot \mid x) = \delta_{Tx}(\cdot), where \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, though nothing can be said about the amount of noise that is tolerable. ==See also==