Absolutely continuous invariant measure

A dynamical system, treated as a space $X$ with a mapping $T : X \rightarrow X$ or a family of mappings $\mathcal{T}$, may have a large number of invariant measures (cf. also Invariant measure). Among them there are invariant measures that are absolutely continuous with respect to some canonical measure on $X$ (cf. also Absolutely continuous measures), such as Lebesgue measure for $X \subset {\bf R} ^ { n }$, Haar measure when $X$ is a topological group, or a product measure when $X$ is a shift space (cf. Shift dynamical system). The importance of absolutely continuous invariant measures is due to a heuristic belief that canonical measures are the ones which represent physical objects.

There is a natural procedure for finding an absolutely continuous invariant measure, by iterating the canonical measure $\mu$. First construct the images of $\mu$ under the mapping $\mu _ { n } = \mu \circ T ^ { - n }$, then take the averages $\nu _ { n } = \sum _ { k = 0 } ^ { n - 1 } \mu _ { k } / n$ and take some weak$\square ^ { * }$ accumulation point. Special properties of the mapping (e.g. its uniform expansion) may be reflected in the properties of the limit measure (absolute continuity). An alternative (dual) way is to iterate the density function with the transfer operator, and use the properties of $T$ to prove a compactness property of a resulting sequence. The existence of an absolutely continuous invariant measure is not granted and is due in many cases to hyperbolic properties of the mapping, such as large derivatives on big sets of points. Once found, the absolutely continuous invariant measure serves via the ergodic theorem to pronounce statements about typical (with respect to the canonical measure) behaviour of the system.

The ergodic theorem says that the long-time behaviour of the system is asymptotically described by the behaviour on ergodic components of the space. The time averages of observables (measurable functions) are then equal to their space averages (integrals). An invariant measure is ergodic if there are no non-trivial invariant sets — if $T ^ { - 1 } A = A$ then either $\mu ( A ) = 0$ or $\mu ( X \backslash A ) = 0$. One can say, imprecisely, that any invariant measure is a combination of invariant ergodic measures.

One calls an invariant measure a Sinai–Bowen–Ruelle measure, or SBR measure, when it is a limit point of the averages of Dirac measures (cf. also Dirac distribution) on the trajectories of points from a set of positive Lebesgue measure:

\begin{equation*} \nu = \operatorname { lim } \sum _ { k = 0 } ^ { n - 1 } \frac { 1 } { n } \delta _ { T ^ { n } x } \end{equation*}

for any $x \in A$ with positive measure. When an SBR measure is absolutely continuous with respect to some natural measure on the space (most often the Lebesgue or Haar measure), then it is said that the system is chaotic or stochastic. When, on the other hand, the SBR measure is concentrated on a finite number of points, then the system is called deterministic (with a periodic attractor). All other systems are commonly called strange or wild. It is widely believed that typically the systems are either stochastic or deterministic (or a combination of them), but there are known examples of strange limit behaviour.

See also Strange attractor; Chaos.

References

[a1]	J. de Vries, "Elements of topological dynamics" , Kluwer Acad. Publ. (1993)
[a2]	R.L. Devaney, "An introduction to chaotic dynamical systems" , Benjamin/Cummings (1986)
[a3]	Yu.I. Neimark, P.S. Landa, "Stochastic and chaotic oscillations" , Kluwer Acad. Publ. (1992) pp. Chap. 2
[a4]	I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982)
[a5]	U. Krengel, "Ergodic theorems" , de Gruyter (1985)