nLab
ergodic decomposition theorem
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Measure and probability theory
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
The ergodic decomposition theorem says that, under some conditions, every invariant measure is a mixture of ergodic ones .

In terms of category theory , the statement can be often expressed in terms of a categorical limit .

Statement of the theorem
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Examples
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Almost sure version
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References
Terence Tao , What’s new? Lecture 9: Ergodicity , blog entry.

Sean Moss, Paolo Perrone , A category-theoretic proof of the ergodic decomposition theorem , Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353 )

Noé Ensarguet, Paolo Perrone , Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

Created on July 15, 2024 at 16:13:57.
See the history of this page for a list of all contributions to it.