---
tag: 256NP6JBEQ
kind: concept
name: Sinai–Ruelle–Bowen measure
msc: ["37D20", "37C40", "37D35"]
sources:
  - url: https://en.wikipedia.org/wiki/Sinai%E2%80%93Ruelle%E2%80%93Bowen_measure
same_as:
  wikidata: Q104867754
  wikipedia: Sinai–Ruelle–Bowen measure
---

## Summary

A Sinai–Ruelle–Bowen measure, or SRB measure, is an [[invariant measure|B232EZ1FMB]] of a smooth dynamical system that is smooth along the expanding directions.
Such measures exist for many [[attractors|7FWX24X5MW]] that have zero volume, where no invariant measure can be absolutely continuous.
They are the natural measures to observe.
Take a uniformly hyperbolic attractor.
For almost every initial point of its basin, in the sense of volume, the time averages equal the SRB space averages.

## Explanations

### L1 human

Run a chaotic system on a computer from a random starting point and record how often the orbit visits each region.
The histogram usually settles down to the same limit for almost every start; that limit is the SRB measure.
On a strange attractor the measure lives on a set of zero volume, so it is not given by a density in space.
Along the stretched directions, however, it looks like length or area with a density.
For [[Arnold's cat map|BSCZN8Y1AN]] the SRB measure is plain area on the torus.