Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Measure-preserving dynamical system

Open this concept in the interactive graph Read the Markdown equivalent

Summary

A measurable dynamical system equipped with a measure invariant under time evolution. For a transformation $T$ , invariance means $\mu(T^{-1}A)=\mu(A)$ for every measurable set $A$ .

Record metadata

Canonical name: Measure-preserving dynamical system
Concept ID: measure_preserving_dynamical_system
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Dynamical systems
Domains: Dynamical systems, Measure, Probability
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Measurable dynamical system

Concept sources

Wikipedia — Measure-preserving dynamical system — Measure-preserving dynamical system · encyclopedic reference · source ID wp-math-measure-preserving-system
nLab — invariant measure — invariant measure · mathematical reference · source ID nlab-invariant-measure
MIT OpenCourseWare — Bits of ergodic theory — Chapter 8: Compressing Stationary Ergodic Sources · university lecture notes · source ID mit-ocw-ergodic-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Dynamical system Measure-preserving dynamical system

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: dyn_system_to_measure_preserving_system
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: invariant measure

Authored explanation

Equip a measurable dynamical system with a chosen measure that is preserved by every evolution map; not every system admits the desired finite invariant measure.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Measure-preserving dynamical system — Measure-preserving dynamical system · encyclopedic reference · source ID wp-math-measure-preserving-system
nLab — invariant measure — invariant measure · mathematical reference · source ID nlab-invariant-measure
MIT OpenCourseWare — Bits of ergodic theory — Chapter 8: Compressing Stationary Ergodic Sources · university lecture notes · source ID mit-ocw-ergodic-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.