Measure Sinai - Ruell - Bowen

The Sinai – Ruelle – Bowen measure , or SRB measure , is a measure on the phase space of a dynamical system that the distribution of the trajectories of typical initial (in the sense of Lebesgue measure) points (possibly from some region) tends to. Moreover, the set of points for which such an aspiration occurs is called the basin of attraction of this measure.

The concept is named after J. G. Sinai , D. Ruelle and R. Bowen , in whose works it was introduced.

Definitions

More precisely, there are two nonequivalent concepts: the definition of the Sinai – Ruel – Bowen measure associated with iterations of typical points (“observable measure”), and its modification associated with iterations of absolutely continuous measures (“natural measure”).

Definition 1 . Measure $\mu$ ${\ displaystyle \ mu}$ is called the (observable) Sinai-Ruelle-Bowen measure if, for the set of starting points $x$ ${\ displaystyle x}$ positive Lebesgue measure, the distribution of orbits converges to $\mu$ ${\ displaystyle \ mu}$ :
${\frac {1}{n}}\sum _{j=0}^{n-1}\delta _{f^{j}(x)}\to \mu ,\quad n\to \infty .\qquad \qquad (*)$ ${\ displaystyle {\ frac {1} {n}} \ sum _ {j = 0} ^ {n-1} \ delta _ {f ^ {j} (x)} \ to \ mu, \ quad n \ to \ infty. \ qquad \ qquad (*)}$
In this case, the set of points x satisfying (*) is called the basin of attraction of the measure $\mu$ ${\ displaystyle \ mu}$ .

Equivalently, this definition can be formulated in terms of time averages :

Definition 1 '. Measure $\mu$ ${\ displaystyle \ mu}$ is called the (observable) Sinai-Ruelle-Bowen measure if, for some set $M$ ${\ displaystyle M}$ positive Lebesgue measures temporary means of any continuous function $\varphi$ ${\ displaystyle \ varphi}$ on $M$ ${\ displaystyle M}$ converge almost everywhere to its integral as $\mu$ ${\ displaystyle \ mu}$
${\frac {1}{n}}\sum _{j=0}^{n-1}\varphi (f^{j}(x)){\xrightarrow[{n\to \infty }]{{\text{a.e. in}}\,\,M}}\int \varphi \,d\mu .\qquad \qquad (**)$ ${\ displaystyle {\ frac {1} {n}} \ sum _ {j = 0} ^ {n-1} \ varphi (f ^ {j} (x)) {\ xrightarrow [{n \ to \ infty} ] {{\ text {ae in}} \, \, M}} \ int \ varphi \, d \ mu. \ qquad \ qquad (**)}$
In this case, the maximum set $M$ ${\ displaystyle M}$ for which (**) holds is called the basin of attraction of the measure $\mu$ ${\ displaystyle \ mu}$ .

In the case of a natural measure, iterations are considered not of the atomic initial measure (or, which is the same, the distribution of an individual orbit), but the averaging of absolutely continuous initial measures:

Definition 2. Measure $\mu$ ${\ displaystyle \ mu}$ is called a (natural) Sinai-Ruelle-Bowen measure if, for some set $M$ ${\ displaystyle M}$ positive Lebesgue measure for any absolutely continuous initial measure m its temporal means converge almost everywhere $\mu$ ${\ displaystyle \ mu}$ :
${\frac {1}{n}}\sum _{j=0}^{n-1}(f^{j})_{*}(m)\to \mu ,\quad n\to \infty .\qquad \qquad (***)$ ${\ displaystyle {\ frac {1} {n}} \ sum _ {j = 0} ^ {n-1} (f ^ {j}) _ {*} (m) \ to \ mu, \ quad n \ to \ infty. \ qquad \ qquad (***)}$
In this case, the maximum measurable set $M$ ${\ displaystyle M}$ for which (***) holds is called the basin of attraction of the measure $\mu$ ${\ displaystyle \ mu}$ .

Literature

Ya.G. Sinai, Gibbs measures in ergodic theory, Advances in Mathematical Sciences , 27 : 4 (1972), 21--69.
R. Bowen. Equilibrium states and ergodic theory of Anosov diffeomorphisms. Springer Lecture Notes in Math. 470 (1975).
D. Ruelle. A measure associated with Axiom A attractors. Amer. J. Math. 98 (1976), pp. 619--654.
L.-S. Young What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108 (2002), pp. 733–754.
M. Blank, L. Bunimovich. Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity , 16 (2003), pp. 387-401.

Measure Sinai - Ruell - Bowen

Definitions

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