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
(…)
Examples
(…)
Almost sure version
(…)
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.